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Задать вопрос экспертам

Консультация № 177617

03.04.2010, 01:16

0.00 руб.

0 1 1

Образование

здравствуйте,уважаемые эксперты.помогите пожалуйста решить задачу:
сторона основания правильной четырехугольной пирамиды равна а,боковая грань составляет с плоскостью основания угол,равный [$945$].найти радиус описанного шара.

заранее благодарен.

Обсуждение

Неизвестный

03.04.2010, 13:40

общий

это ответ

Здравствуйте, G-buck.

SABCD - пирамида
ABCD - квадрат со стороной a
H - высота пирамиды
O - пересечение диагоналей ABCD
H=OM*tg([$945$])=(a/2)*tg([$945$])

Центр описанного шара (сферы) - точка пересечения
плоскостей, проведенных через середины боковых ребер пирамиды, перпендикулярно им.
Т.к. пирамида правильная OSA=OSB=OSC=OSD
Рассмотрим треугольник OSD
Точка пересечения перпендикуляра, проведенного через середину SD с OS - центр описанного шара
OS=H
R-радиус описанного шара
OD=a*[$8730$]2/2 - как половина диагонали квадрата
p=SD=[$8730$](OS²+OD²)=(a/2)*[$8730$](tg²([$945$])+2)
cos([$946$])=H/p=(p/2)/R
Получим
R=p²/(2*H)=(a/4)*(tg²([$945$])+2)/tg([$945$])

Ответ: (a/4)*(tg²([$945$])+2)/tg([$945$])

5

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