Question

стороны основания треугольной пирамиды равны 15,16 и 17. все боковые ребра наклонены к плоскости основания под углом 45. найдите объем пирамиды.

Answer 1

V=S*h/3

S осн= sqrt(p(p-a)(p-b)(p-c));

sqrt- корень

p- полупериметр

a,b,c-стороны треугольника

R=a*b*c/4*S

в нашем случае R=h,

теперь найдем объем подставив данные

Answer 2

Given Information

We are given that the sides of the triangular pyramid are equal to 15, 16, and 17 units. All the lateral edges are inclined to the base plane at an angle of 45 degrees.

Finding the Volume of the Pyramid

To find the volume of the pyramid, we can use the formula:

Volume = (1/3) * Base Area * Height

The base area can be calculated using Heron's formula, which states that the area of a triangle with sides a, b, and c is given by:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle, given by:

s = (a + b + c) / 2

In this case, the sides of the base triangle are 15, 16, and 17 units. Let's calculate the base area:

s = (15 + 16 + 17) / 2 = 48 / 2 = 24

Area = sqrt(24 * (24 - 15) * (24 - 16) * (24 - 17)) = sqrt(24 * 9 * 8 * 7) = sqrt(12096) ≈ 110.04

Now, we need to find the height of the pyramid. Since all the lateral edges are inclined to the base plane at an angle of 45 degrees, we can consider the height as the perpendicular distance from the apex of the pyramid to the base plane.

To find the height, we can use the Pythagorean theorem. Let's consider the right triangle formed by the height, the lateral edge, and half of the base side. The hypotenuse of this triangle is the lateral edge, and one of the legs is half of the base side.

Let's calculate the length of the height using the Pythagorean theorem:

Height = sqrt(lateral edge^2 - (base side / 2)^2)

Height = sqrt(17^2 - (15 / 2)^2) = sqrt(289 - 112.5) = sqrt(176.5) ≈ 13.28

Now, we can calculate the volume of the pyramid using the formula:

Volume = (1/3) * Base Area * Height

Volume = (1/3) * 110.04 * 13.28 ≈ 484.91 cubic units

Therefore, the volume of the given triangular pyramid is approximately