Question

Дано: Правильная четырехугольная пирамида Боковое ребро образует с высотой угол 45 Сторона основания 2 Найти: Полную площадь

Answer 1

Дано: МАВСД правильная пирамида. АВ=2, <MAC=45°
найти: Sполн.пов

решение.
Sполн.пов=Sбок+Sосн
Sбок=Росн*ha, ha-апофема
Sосн=а²

АВСД - квадрат. найдем диагональ АС по теореме Пифагора:
АС²=АВ²+ВС². АС=2√2
рассмотрим ΔМАО:
 (О- точка пересечения диагоналей квадрата-основания пирамиды)
<MAO=45°,
AO=2√2/2, AO=√2. ΔMAO - прямоугольный равнобедренный, ⇒МО=√2
МК-апофема.
рассмотрим ΔМОК: <MOK=90°(MO-высота пирамиды)
ОК=2:2, ОК=1
найдем МК по тереме Пифагора:
МК²=МО²+ОК², МК=√3
Sполн.пов=(4*2*√3)+2²=8√3+4
Sполн.пов=8√3+4

Answer 2

Given Information:

We are given a right quadrilateral pyramid where the lateral edge forms a 45-degree angle with the height, and the side length of the base is 2.

Approach:

To find the total surface area of the pyramid, we need to calculate the areas of the base and the four triangular faces.

Calculation:

1. Base Area: - The base of the pyramid is a quadrilateral. - Since it is a right quadrilateral pyramid, the base is a square. - The side length of the base is given as 2. - The area of a square is calculated by squaring the length of one side. - Therefore, the area of the base is 2 * 2 = 4.

2. Lateral Surface Area: - The pyramid has four triangular faces. - The lateral surface area of a pyramid is calculated by summing the areas of all the triangular faces. - Each triangular face has a base equal to the side length of the base and a height equal to the height of the pyramid. - The height of the pyramid is not given explicitly, but we know that the lateral edge forms a 45-degree angle with the height. - Since the lateral edge forms a 45-degree angle with the height, we can use trigonometry to find the height. - The height can be calculated as the product of the lateral edge length and the sine of the angle formed with the height. - The sine of 45 degrees is 1/sqrt(2) or approximately 0.7071. - Therefore, the height of the pyramid is 2 * 0.7071 = 1.4142. - Now, we can calculate the area of each triangular face using the formula: (1/2) * base * height. - The base of each triangular face is 2, and the height is 1.4142. - Therefore, the area of each triangular face is (1/2) * 2 * 1.4142 = 1.4142. - Since there are four triangular faces, the total lateral surface area is 4 * 1.4142 = 5.6568.

3. Total Surface Area: - The total surface area of the pyramid is the sum of the base area and the lateral surface area. - The base area is 4, and the lateral surface area is 5.6568. - Therefore, the total surface area of the pyramid is 4 + 5.6568 = 9.6568.

Answer:

The total surface area of the given pyramid is approximately 9.6568 square units.