Question

В основании пирамиды лежит прямоугольный треугольник с катетами 8 и 15. Все двугранные углы пирамиды при сторонах основания равны. Найдите площадь полной поверхности пирамиды, если высота пирамиды равна .

Answer 1

Гипотенуза основания равна √(8² + 15²) = √289 = 17.

Если все двугранные углы пирамиды при сторонах основания равны, то проекция вершины пирамиды на основание - это центр вписанной в основание окружности.

Радиус этой окружности r = (a + b - c)/2 = (8 + 15 - 17)/2 = 5/2.

Тангенс угла наклона боковых граней к основанию равен:

tg α = H/r = 3√3/(5/2) = 6√3/5.

cos α = 1/(1+ tg²α) = 5/√133.

Площадь основания равна So = (1/2)*8*15 = 60.

Площадь боковой поверхности равна:

Sбок = So/cos α = 60/(5/√133) = 12√133.

Площадь полной поверхности пирамиды равна:

S = Sо + Sбок = 60 + 12√133 = 12(5 + √133) кв.ед.

Answer 2

Finding the Surface Area of the Pyramid

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

1. Area of the triangular faces:

Since all the triangular faces of the pyramid have equal angles, we can calculate the area of one triangular face and then multiply it by the number of faces.

To calculate the area of a triangle, we can use the formula: Area = (1/2) * base * height.

In this case, the base of the triangle is one of the sides of the rectangular triangle, which is 8 units. The height of the triangle can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using the given sides of the rectangular triangle (8 and 15), we can find the height as follows:

height = sqrt(15^2 - 8^2) = sqrt(225 - 64) = sqrt(161).

Now we can calculate the area of one triangular face:

Area = (1/2) * 8 * sqrt(161).

To find the total area of all the triangular faces, we need to multiply this area by the number of faces. Since the pyramid has four triangular faces, the total area of the triangular faces is:

Total area of triangular faces = 4 * (1/2) * 8 * sqrt(161).

2. Area of the base:

The base of the pyramid is a rectangle with sides 8 and 15. The area of a rectangle is calculated by multiplying its length and width, so the area of the base is:

Area of the base = 8 * 15.

3. Total surface area of the pyramid:

To find the total surface area of the pyramid, we need to add the areas of the triangular faces and the area of the base:

Total surface area = Total area of triangular faces + Area of the base.

Let's calculate the values:

Total area of triangular faces = 4 * (1/2) * 8 * sqrt(161).

Area of the base = 8 * 15.

Total surface area = 4 * (1/2) * 8 * sqrt(161) + 8 * 15.

Now we can calculate the total surface area of the pyramid.