Question

Задача. в правильной четырехугольной пирамиде высота 4 корней из 3, а диагональ основания 8 корней из 2. найти площадь боковой поверхности пирамиды

Answer 1

Площадь боковой поверхности правильной пирамиды - периметр основания умноженный на высоту.
Высота - 4√3.
Основание правильной четырехугольной пирамиды - квадрат.
Диагональ квадрата - 8√2. По т. Пифагора находим сторону квадрата:
2а²=(8√2)²
а=6;
Периметр - 8*4=32;
Площадь боковой поверхности - 32*4√3=128√3 ед².

Answer 2

Task Analysis

To find the lateral surface area of a regular quadrilateral pyramid, we are given the height (4√3) and the base diagonal (8√2) of the pyramid. We need to calculate the lateral surface area of the pyramid.

Solution

The lateral surface area (S) of a regular quadrilateral pyramid can be calculated using the formula: S = 1/2 * Perimeter of the base * Slant height

First, let's calculate the perimeter of the base and the slant height.

The perimeter of the base (P) can be calculated using the formula: P = 4 * side length

The slant height (l) can be calculated using the formula: l = √(h^2 + (P/2)^2)

Where: - h = height of the pyramid - P = perimeter of the base

Calculations

1. Calculate the perimeter of the base (P): - Given: Base diagonal = 8√2 - Using the formula for the diagonal of a square: diagonal = side length * √2 - Therefore, side length = diagonal / √2

Substituting the given value: side length = 8√2 / √2 = 8

Now, calculate the perimeter: P = 4 * side length = 4 * 8 = 32

The perimeter of the base (P) is 32.

2. Calculate the slant height (l): - Given: Height (h) = 4√3 - Using the formula for the slant height: l = √(h^2 + (P/2)^2) l = √((4√3)^2 + (32/2)^2) l = √(48 + 256) l = √304

The slant height (l) is √304.

3. Calculate the lateral surface area (S): - Using the formula: S = 1/2 * P * l S = 1/2 * 32 * √304 S = 16 * √304

Therefore, the lateral surface area of the pyramid is 16√304.

Conclusion

The lateral surface area of the given regular quadrilateral pyramid is 16√304.