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Номер 428, страница 114 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 4. Цилиндр, конус и шар. Параграф 3. Сфера, дополнительные задачи - номер 428, страница 114.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744843 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "114" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-428" "field_display_title" => "428" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "89" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "89" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:2 [ 0 => App\Models\Branch {#1077 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "89" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Book {#1082 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} …42 ] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080} "field_page_start" => "89" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1081} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1118 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 744391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сфера, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_page_start" => "112" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Book {#1122 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Branch {#1158 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сфера, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119} "field_page_start" => "112" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1121} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:6 [ 0 => App\Models\Element {#1178 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 939077 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1179 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744843" "type" => "task" ] "text" => "<p> <b>428.</b> Около сферы радиуса R описана правильная четырёхугольная пирамида, плоский угол при вершине которой равен α. </p> <p> а) Найдите площадь боковой поверхности пирамиды. </p> <p> б) Вычислите эту площадь при <span class="long"> R = 5 см, </span> <span class="long"> α = 60°.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "1559" "height" => 250 "path" => "/media/geometrija_10/atanasjan-u23/0-00/428-1.webp?ts=1739436639" ] ] ] #original: array:7 [ "id" => 939077 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "744843" "type" => "task" ] "text" => "<p> <b>428.</b> Около сферы радиуса R описана правильная четырёхугольная пирамида, плоский угол при вершине которой равен α. </p> <p> а) Найдите площадь боковой поверхности пирамиды. </p> <p> б) Вычислите эту площадь при <span class="long"> R = 5 см, </span> <span class="long"> α = 60°.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "1559" "height" => 250 "path" => "/media/geometrija_10/atanasjan-u23/0-00/428-1.webp?ts=1739436639" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1209 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Edition {#1210 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "428-1.png" "alt" => null "width" => "694" "height" => 11777 "path" => "/media/geometrija_10/atanasjan-u23/2-00/428-1.webp?ts=1739441295" ] 1 => array:5 [ "name" => "428-2.png" "alt" => null "width" => "694" "height" => 1371 "path" => "/media/geometrija_10/atanasjan-u23/2-00/428-2.webp?ts=1739441295" ] ] ] #original: array:6 [ "id" => 941736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219} "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "428-1.png" "alt" => null "width" => "694" "height" => 11777 "path" => "/media/geometrija_10/atanasjan-u23/2-00/428-1.webp?ts=1739441295" ] 1 => array:5 [ "name" => "428-2.png" "alt" => null "width" => "694" "height" => 1371 "path" => "/media/geometrija_10/atanasjan-u23/2-00/428-2.webp?ts=1739441295" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1201 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Edition {#1202 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "1358" "height" => 173 "path" => "/media/geometrija_10/atanasjan-u23/3-00/428-1.webp?ts=1739436880" ] 1 => array:5 [ "name" => "428-2.jpg" "alt" => null "width" => "1358" "height" => 1994 "path" => "/media/geometrija_10/atanasjan-u23/3-00/428-2.webp?ts=1739436880" ] 2 => array:5 [ "name" => "428-3.jpg" "alt" => null "width" => "1358" "height" => 314 "path" => "/media/geometrija_10/atanasjan-u23/3-00/428-3.webp?ts=1739436880" ] ] ] #original: array:6 [ "id" => 941568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211} "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "1358" "height" => 173 "path" => "/media/geometrija_10/atanasjan-u23/3-00/428-1.webp?ts=1739436880" ] 1 => array:5 [ "name" => "428-2.jpg" "alt" => null "width" => "1358" "height" => 1994 "path" => "/media/geometrija_10/atanasjan-u23/3-00/428-2.webp?ts=1739436880" ] 2 => array:5 [ "name" => "428-3.jpg" "alt" => null "width" => "1358" "height" => 314 "path" => "/media/geometrija_10/atanasjan-u23/3-00/428-3.webp?ts=1739436880" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1185 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941072 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Edition {#1186 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "1372" "height" => 979 "path" => "/media/geometrija_10/atanasjan-u23/4-00/428-1.webp?ts=1739436566" ] 1 => array:5 [ "name" => "428-2.jpg" "alt" => null "width" => "1372" "height" => 2158 "path" => "/media/geometrija_10/atanasjan-u23/4-00/428-2.webp?ts=1739436566" ] 2 => array:5 [ "name" => "428-3.jpg" "alt" => null "width" => "1372" "height" => 882 "path" => "/media/geometrija_10/atanasjan-u23/4-00/428-3.webp?ts=1739436566" ] ] ] #original: array:6 [ "id" => 941072 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "1372" "height" => 979 "path" => "/media/geometrija_10/atanasjan-u23/4-00/428-1.webp?ts=1739436566" ] 1 => array:5 [ "name" => "428-2.jpg" "alt" => null "width" => "1372" "height" => 2158 "path" => "/media/geometrija_10/atanasjan-u23/4-00/428-2.webp?ts=1739436566" ] 2 => array:5 [ "name" => "428-3.jpg" "alt" => null "width" => "1372" "height" => 882 "path" => "/media/geometrija_10/atanasjan-u23/4-00/428-3.webp?ts=1739436566" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1193 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 941465 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1194 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "499" "height" => 1031 "path" => "/media/geometrija_10/atanasjan-u23/5-00/428-1.webp?ts=1739436631" ] ] ] #original: array:6 [ "id" => 941465 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "744843" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "428-1.jpg" "alt" => null "width" => "499" "height" => 1031 "path" => "/media/geometrija_10/atanasjan-u23/5-00/428-1.webp?ts=1739436631" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1217 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1352156 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Edition {#1218 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744843" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Пусть дана правильная четырехугольная пирамида $SABCD$, где $ABCD$ – квадратное основание, а $S$ – вершина. Пусть $O$ – центр основания, тогда $SO$ – высота пирамиды $H$. Пусть $a$ – сторона основания ($AB=BC=CD=DA=a$), а $h_a$ – апофема пирамиды (высота боковой грани, например, $SM \perp BC$, где $M$ – середина $BC$).</p><p>Площадь боковой поверхности пирамиды $S_{бок}$ равна сумме площадей четырех одинаковых боковых граней: $S_{бок} = 4 \cdot S_{\triangle SBC} = 4 \cdot \frac{1}{2} a \cdot h_a = 2ah_a$.</p><p>Рассмотрим боковую грань – равнобедренный треугольник $SBC$. Угол при вершине $\angle BSC = \alpha$. Апофема $SM$ является также биссектрисой, поэтому $\angle BSM = \frac{\alpha}{2}$. В прямоугольном треугольнике $SMB$ имеем: $BM = \frac{a}{2}$. $\tan(\frac{\alpha}{2}) = \frac{BM}{SM} = \frac{a/2}{h_a}$.</p><p>Рассмотрим сечение пирамиды, проходящее через апофемы противоположных граней, например, $SM$ и $SN$ (где $N$ - середина $AD$). Это сечение представляет собой равнобедренный треугольник $MSN$. $SO$ - его высота, $MN = a$. Вписанная в пирамиду сфера касается основания в точке $O$ и боковых граней в точках, лежащих на апофемах. Центр сферы $O_s$ лежит на высоте $SO$, и расстояние от $O_s$ до плоскости основания равно радиусу $R$, то есть $O_sO = R$.</p><p>Рассмотрим прямоугольный треугольник $SOM$. Угол $\angle SMO = \beta$ – это двугранный угол при ребре основания. В этом треугольнике: $OM = \frac{a}{2}$. $\cos(\beta) = \frac{OM}{SM} = \frac{a/2}{h_a}$.</p><p>Сравнивая выражения для $\frac{a/2}{h_a}$ из треугольников $SMB$ и $SOM$, получаем важную связь между плоским углом при вершине $\alpha$ и двугранным углом при основании $\beta$: $\cos(\beta) = \tan(\frac{\alpha}{2})$.</p><p>Теперь свяжем параметры пирамиды с радиусом вписанной сферы $R$. В сечении $MSN$ вписанная сфера выглядит как окружность радиуса $R$, вписанная в треугольник $MSN$. Центр этой окружности $O_s$ лежит на высоте $SO$. В прямоугольном треугольнике $SOM$ проведем биссектрису угла $\beta$, она пройдет через центр вписанной окружности $O_s$. Рассмотрим прямоугольный треугольник $O_sOM$, образованный катетами $O_sO=R$ и $OM=\frac{a}{2}$. Угол $\angle O_sMO = \frac{\beta}{2}$. Из этого треугольника: $\tan(\frac{\beta}{2}) = \frac{O_sO}{OM} = \frac{R}{a/2} = \frac{2R}{a}$. Отсюда выразим сторону основания $a$: $a = \frac{2R}{\tan(\frac{\beta}{2})} = 2R \cot(\frac{\beta}{2})$.</p><p>Теперь выразим площадь боковой поверхности через $R$ и $\beta$: $S_{бок} = 2ah_a = 2a \cdot \frac{a/2}{\cos\beta} = \frac{a^2}{\cos\beta}$. Подставляем выражение для $a$: $S_{бок} = \frac{(2R \cot(\frac{\beta}{2}))^2}{\cos\beta} = \frac{4R^2 \cot^2(\frac{\beta}{2})}{\cos\beta}$.</p><p>Используем тригонометрическую формулу для половинного угла: $\cot^2(\frac{\beta}{2}) = \frac{1+\cos\beta}{1-\cos\beta}$. $S_{бок} = \frac{4R^2}{\cos\beta} \cdot \frac{1+\cos\beta}{1-\cos\beta}$.</p><p>Наконец, заменим $\cos\beta$ на $\tan(\frac{\alpha}{2})$: $S_{бок} = \frac{4R^2}{\tan(\frac{\alpha}{2})} \cdot \frac{1+\tan(\frac{\alpha}{2})}{1-\tan(\frac{\alpha}{2})} = \frac{4R^2(1+\tan(\frac{\alpha}{2}))}{\tan(\frac{\alpha}{2})(1-\tan(\frac{\alpha}{2}))}$.</p><p>Ответ: $S_{бок} = \frac{4R^2(1+\tan(\frac{\alpha}{2}))}{\tan(\frac{\alpha}{2}) - \tan^2(\frac{\alpha}{2})}$</p><p><strong>б)</strong></p><p>Вычислим площадь при $R=5$ см и $\alpha=60^\circ$. Сначала найдем значение $\tan(\frac{\alpha}{2})$: $\frac{\alpha}{2} = \frac{60^\circ}{2} = 30^\circ$. $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.</p><p>Теперь подставим числовые значения в полученную формулу: $S_{бок} = \frac{4 \cdot 5^2 \cdot (1+\frac{1}{\sqrt{3}})}{\frac{1}{\sqrt{3}} \cdot (1-\frac{1}{\sqrt{3}})} = \frac{100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}-1}{\sqrt{3}}} = \frac{100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{\sqrt{3}-1}{3}}$.</p><p>$S_{бок} = 100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}} \cdot \frac{3}{\sqrt{3}-1} = 100 \cdot \frac{3(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3}-1)}$. $S_{бок} = 100 \cdot \frac{\sqrt{3}\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3}-1)} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}-1}$.</p><p>Домножим числитель и знаменатель на сопряженное выражение $(\sqrt{3}+1)$: $S_{бок} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)^2}{(\sqrt{3})^2-1^2}$. $S_{бок} = 100 \cdot \frac{\sqrt{3}(3+2\sqrt{3}+1)}{3-1} = 100 \cdot \frac{\sqrt{3}(4+2\sqrt{3})}{2}$. $S_{бок} = 50 \cdot \sqrt{3}(4+2\sqrt{3}) = 50 \cdot (4\sqrt{3} + 2\cdot3) = 50 \cdot (6+4\sqrt{3})$. $S_{бок} = 100(3+2\sqrt{3})$ см$^2$.</p><p>Ответ: $100(3+2\sqrt{3})$ см$^2$</p>" ] #original: array:6 [ "id" => 1352156 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227} "task" => array:2 [ "refs" => "744843" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Пусть дана правильная четырехугольная пирамида $SABCD$, где $ABCD$ – квадратное основание, а $S$ – вершина. Пусть $O$ – центр основания, тогда $SO$ – высота пирамиды $H$. Пусть $a$ – сторона основания ($AB=BC=CD=DA=a$), а $h_a$ – апофема пирамиды (высота боковой грани, например, $SM \perp BC$, где $M$ – середина $BC$).</p><p>Площадь боковой поверхности пирамиды $S_{бок}$ равна сумме площадей четырех одинаковых боковых граней: $S_{бок} = 4 \cdot S_{\triangle SBC} = 4 \cdot \frac{1}{2} a \cdot h_a = 2ah_a$.</p><p>Рассмотрим боковую грань – равнобедренный треугольник $SBC$. Угол при вершине $\angle BSC = \alpha$. Апофема $SM$ является также биссектрисой, поэтому $\angle BSM = \frac{\alpha}{2}$. В прямоугольном треугольнике $SMB$ имеем: $BM = \frac{a}{2}$. $\tan(\frac{\alpha}{2}) = \frac{BM}{SM} = \frac{a/2}{h_a}$.</p><p>Рассмотрим сечение пирамиды, проходящее через апофемы противоположных граней, например, $SM$ и $SN$ (где $N$ - середина $AD$). Это сечение представляет собой равнобедренный треугольник $MSN$. $SO$ - его высота, $MN = a$. Вписанная в пирамиду сфера касается основания в точке $O$ и боковых граней в точках, лежащих на апофемах. Центр сферы $O_s$ лежит на высоте $SO$, и расстояние от $O_s$ до плоскости основания равно радиусу $R$, то есть $O_sO = R$.</p><p>Рассмотрим прямоугольный треугольник $SOM$. Угол $\angle SMO = \beta$ – это двугранный угол при ребре основания. В этом треугольнике: $OM = \frac{a}{2}$. $\cos(\beta) = \frac{OM}{SM} = \frac{a/2}{h_a}$.</p><p>Сравнивая выражения для $\frac{a/2}{h_a}$ из треугольников $SMB$ и $SOM$, получаем важную связь между плоским углом при вершине $\alpha$ и двугранным углом при основании $\beta$: $\cos(\beta) = \tan(\frac{\alpha}{2})$.</p><p>Теперь свяжем параметры пирамиды с радиусом вписанной сферы $R$. В сечении $MSN$ вписанная сфера выглядит как окружность радиуса $R$, вписанная в треугольник $MSN$. Центр этой окружности $O_s$ лежит на высоте $SO$. В прямоугольном треугольнике $SOM$ проведем биссектрису угла $\beta$, она пройдет через центр вписанной окружности $O_s$. Рассмотрим прямоугольный треугольник $O_sOM$, образованный катетами $O_sO=R$ и $OM=\frac{a}{2}$. Угол $\angle O_sMO = \frac{\beta}{2}$. Из этого треугольника: $\tan(\frac{\beta}{2}) = \frac{O_sO}{OM} = \frac{R}{a/2} = \frac{2R}{a}$. Отсюда выразим сторону основания $a$: $a = \frac{2R}{\tan(\frac{\beta}{2})} = 2R \cot(\frac{\beta}{2})$.</p><p>Теперь выразим площадь боковой поверхности через $R$ и $\beta$: $S_{бок} = 2ah_a = 2a \cdot \frac{a/2}{\cos\beta} = \frac{a^2}{\cos\beta}$. Подставляем выражение для $a$: $S_{бок} = \frac{(2R \cot(\frac{\beta}{2}))^2}{\cos\beta} = \frac{4R^2 \cot^2(\frac{\beta}{2})}{\cos\beta}$.</p><p>Используем тригонометрическую формулу для половинного угла: $\cot^2(\frac{\beta}{2}) = \frac{1+\cos\beta}{1-\cos\beta}$. $S_{бок} = \frac{4R^2}{\cos\beta} \cdot \frac{1+\cos\beta}{1-\cos\beta}$.</p><p>Наконец, заменим $\cos\beta$ на $\tan(\frac{\alpha}{2})$: $S_{бок} = \frac{4R^2}{\tan(\frac{\alpha}{2})} \cdot \frac{1+\tan(\frac{\alpha}{2})}{1-\tan(\frac{\alpha}{2})} = \frac{4R^2(1+\tan(\frac{\alpha}{2}))}{\tan(\frac{\alpha}{2})(1-\tan(\frac{\alpha}{2}))}$.</p><p>Ответ: $S_{бок} = \frac{4R^2(1+\tan(\frac{\alpha}{2}))}{\tan(\frac{\alpha}{2}) - \tan^2(\frac{\alpha}{2})}$</p><p><strong>б)</strong></p><p>Вычислим площадь при $R=5$ см и $\alpha=60^\circ$. Сначала найдем значение $\tan(\frac{\alpha}{2})$: $\frac{\alpha}{2} = \frac{60^\circ}{2} = 30^\circ$. $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.</p><p>Теперь подставим числовые значения в полученную формулу: $S_{бок} = \frac{4 \cdot 5^2 \cdot (1+\frac{1}{\sqrt{3}})}{\frac{1}{\sqrt{3}} \cdot (1-\frac{1}{\sqrt{3}})} = \frac{100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}-1}{\sqrt{3}}} = \frac{100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{\sqrt{3}-1}{3}}$.</p><p>$S_{бок} = 100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}} \cdot \frac{3}{\sqrt{3}-1} = 100 \cdot \frac{3(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3}-1)}$. $S_{бок} = 100 \cdot \frac{\sqrt{3}\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3}-1)} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}-1}$.</p><p>Домножим числитель и знаменатель на сопряженное выражение $(\sqrt{3}+1)$: $S_{бок} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)^2}{(\sqrt{3})^2-1^2}$. $S_{бок} = 100 \cdot \frac{\sqrt{3}(3+2\sqrt{3}+1)}{3-1} = 100 \cdot \frac{\sqrt{3}(4+2\sqrt{3})}{2}$. $S_{бок} = 50 \cdot \sqrt{3}(4+2\sqrt{3}) = 50 \cdot (4\sqrt{3} + 2\cdot3) = 50 \cdot (6+4\sqrt{3})$. $S_{бок} = 100(3+2\sqrt{3})$ см$^2$.</p><p>Ответ: $100(3+2\sqrt{3})$ см$^2$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "744844" "type" => "task" ] "previous" => array:2 [ "refs" => "744842" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Book {#1176 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1165 #items: array:1 [ 0 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:2 [ 0 => App\Models\Term {#1174 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1172 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171 #items: array:1 [ 0 => App\Models\Term {#1173 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:5 [ 0 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1170 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1168 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1225 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1226 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Term {#1230 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Term {#1232 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1234 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1235 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1236 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Term {#1238 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Term {#1240 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Term {#1242 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1243 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1244 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1245 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1246 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Term {#1248 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1165} "field_class" => Illuminate\Database\Eloquent\Collection {#1175} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171} "field_author" => Illuminate\Database\Eloquent\Collection {#1169} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1228} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229} "field_country" => Illuminate\Database\Eloquent\Collection {#1231} "field_city" => Illuminate\Database\Eloquent\Collection {#1233} "field_series" => Illuminate\Database\Eloquent\Collection {#1235} "field_umk" => 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№428 (с. 114)
Условие. №428 (с. 114)
скриншот условия
428. Около сферы радиуса R описана правильная четырёхугольная пирамида, плоский угол при вершине которой равен α.
а) Найдите площадь боковой поверхности пирамиды.
б) Вычислите эту площадь при R = 5 см, α = 60°.
Решение 2. №428 (с. 114)
Решение 3. №428 (с. 114)
Решение 4. №428 (с. 114)
Решение 5. №428 (с. 114)
Решение 6. №428 (с. 114)
а)
Пусть дана правильная четырехугольная пирамида $SABCD$, где $ABCD$ – квадратное основание, а $S$ – вершина. Пусть $O$ – центр основания, тогда $SO$ – высота пирамиды $H$. Пусть $a$ – сторона основания ($AB=BC=CD=DA=a$), а $h_a$ – апофема пирамиды (высота боковой грани, например, $SM \perp BC$, где $M$ – середина $BC$).
Площадь боковой поверхности пирамиды $S_{бок}$ равна сумме площадей четырех одинаковых боковых граней: $S_{бок} = 4 \cdot S_{\triangle SBC} = 4 \cdot \frac{1}{2} a \cdot h_a = 2ah_a$.
Рассмотрим боковую грань – равнобедренный треугольник $SBC$. Угол при вершине $\angle BSC = \alpha$. Апофема $SM$ является также биссектрисой, поэтому $\angle BSM = \frac{\alpha}{2}$. В прямоугольном треугольнике $SMB$ имеем: $BM = \frac{a}{2}$. $\tan(\frac{\alpha}{2}) = \frac{BM}{SM} = \frac{a/2}{h_a}$.
Рассмотрим сечение пирамиды, проходящее через апофемы противоположных граней, например, $SM$ и $SN$ (где $N$ - середина $AD$). Это сечение представляет собой равнобедренный треугольник $MSN$. $SO$ - его высота, $MN = a$. Вписанная в пирамиду сфера касается основания в точке $O$ и боковых граней в точках, лежащих на апофемах. Центр сферы $O_s$ лежит на высоте $SO$, и расстояние от $O_s$ до плоскости основания равно радиусу $R$, то есть $O_sO = R$.
Рассмотрим прямоугольный треугольник $SOM$. Угол $\angle SMO = \beta$ – это двугранный угол при ребре основания. В этом треугольнике: $OM = \frac{a}{2}$. $\cos(\beta) = \frac{OM}{SM} = \frac{a/2}{h_a}$.
Сравнивая выражения для $\frac{a/2}{h_a}$ из треугольников $SMB$ и $SOM$, получаем важную связь между плоским углом при вершине $\alpha$ и двугранным углом при основании $\beta$: $\cos(\beta) = \tan(\frac{\alpha}{2})$.
Теперь свяжем параметры пирамиды с радиусом вписанной сферы $R$. В сечении $MSN$ вписанная сфера выглядит как окружность радиуса $R$, вписанная в треугольник $MSN$. Центр этой окружности $O_s$ лежит на высоте $SO$. В прямоугольном треугольнике $SOM$ проведем биссектрису угла $\beta$, она пройдет через центр вписанной окружности $O_s$. Рассмотрим прямоугольный треугольник $O_sOM$, образованный катетами $O_sO=R$ и $OM=\frac{a}{2}$. Угол $\angle O_sMO = \frac{\beta}{2}$. Из этого треугольника: $\tan(\frac{\beta}{2}) = \frac{O_sO}{OM} = \frac{R}{a/2} = \frac{2R}{a}$. Отсюда выразим сторону основания $a$: $a = \frac{2R}{\tan(\frac{\beta}{2})} = 2R \cot(\frac{\beta}{2})$.
Теперь выразим площадь боковой поверхности через $R$ и $\beta$: $S_{бок} = 2ah_a = 2a \cdot \frac{a/2}{\cos\beta} = \frac{a^2}{\cos\beta}$. Подставляем выражение для $a$: $S_{бок} = \frac{(2R \cot(\frac{\beta}{2}))^2}{\cos\beta} = \frac{4R^2 \cot^2(\frac{\beta}{2})}{\cos\beta}$.
Используем тригонометрическую формулу для половинного угла: $\cot^2(\frac{\beta}{2}) = \frac{1+\cos\beta}{1-\cos\beta}$. $S_{бок} = \frac{4R^2}{\cos\beta} \cdot \frac{1+\cos\beta}{1-\cos\beta}$.
Наконец, заменим $\cos\beta$ на $\tan(\frac{\alpha}{2})$: $S_{бок} = \frac{4R^2}{\tan(\frac{\alpha}{2})} \cdot \frac{1+\tan(\frac{\alpha}{2})}{1-\tan(\frac{\alpha}{2})} = \frac{4R^2(1+\tan(\frac{\alpha}{2}))}{\tan(\frac{\alpha}{2})(1-\tan(\frac{\alpha}{2}))}$.
Ответ: $S_{бок} = \frac{4R^2(1+\tan(\frac{\alpha}{2}))}{\tan(\frac{\alpha}{2}) - \tan^2(\frac{\alpha}{2})}$
б)
Вычислим площадь при $R=5$ см и $\alpha=60^\circ$. Сначала найдем значение $\tan(\frac{\alpha}{2})$: $\frac{\alpha}{2} = \frac{60^\circ}{2} = 30^\circ$. $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.
Теперь подставим числовые значения в полученную формулу: $S_{бок} = \frac{4 \cdot 5^2 \cdot (1+\frac{1}{\sqrt{3}})}{\frac{1}{\sqrt{3}} \cdot (1-\frac{1}{\sqrt{3}})} = \frac{100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}-1}{\sqrt{3}}} = \frac{100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}}}{\frac{\sqrt{3}-1}{3}}$.
$S_{бок} = 100 \cdot \frac{\sqrt{3}+1}{\sqrt{3}} \cdot \frac{3}{\sqrt{3}-1} = 100 \cdot \frac{3(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3}-1)}$. $S_{бок} = 100 \cdot \frac{\sqrt{3}\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}(\sqrt{3}-1)} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}-1}$.
Домножим числитель и знаменатель на сопряженное выражение $(\sqrt{3}+1)$: $S_{бок} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = 100 \cdot \frac{\sqrt{3}(\sqrt{3}+1)^2}{(\sqrt{3})^2-1^2}$. $S_{бок} = 100 \cdot \frac{\sqrt{3}(3+2\sqrt{3}+1)}{3-1} = 100 \cdot \frac{\sqrt{3}(4+2\sqrt{3})}{2}$. $S_{бок} = 50 \cdot \sqrt{3}(4+2\sqrt{3}) = 50 \cdot (4\sqrt{3} + 2\cdot3) = 50 \cdot (6+4\sqrt{3})$. $S_{бок} = 100(3+2\sqrt{3})$ см$^2$.
Ответ: $100(3+2\sqrt{3})$ см$^2$
Другие задания:
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ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 10-11 класс, для упражнения номер 428 расположенного на странице 114 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №428 (с. 114), авторов: Атанасян (Левон Сергеевич), Бутузов (Валентин Фёдорович), Кадомцев (Сергей Борисович), Позняк (Эдуард Генрихович), Киселёва (Людмила Сергеевна), ФГОС (старый) базовый и углублённый уровень обучения учебного пособия издательства Просвещение.