One of the bases of the truncated pyramid is an isosceles triangle with a base of 6 cm and a lateral side of 5 cm.

One of the bases of the truncated pyramid is an isosceles triangle with a base of 6 cm and a lateral side of 5 cm. The perimeter of the second base is 32 cm. Find the volume of the pyramid if its height is 4 cm

In an isosceles triangle, the sides are equal, which means the perimeter of one of the bases of this truncated pyramid:

Р1 = 6 + 5 + 5 = 16 cm.

The bases of the truncated pyramid are similar triangles, knowing their perimeter, we find the coefficient of similarity:

k = P2 / P1 = 32/16 = 2.

Knowing that the sides of the smaller base are 5 cm and 6 cm, we find the sides of the larger base:

5 * 2 = 10 cm;

6 * 2 = 12 cm.

The area of ​​the base of the pyramid can be determined by Heron’s formula:

S1 = √8 * (8 – 5) * (8 – 5) * (8 – 6) = √8 * 3 * 3 * 2 = 12 cm2 – the area of ​​the smaller base;

S2 = √16 * (16 – 10) * (16 – 10) * (16 – 12) = √16 * 6 * 6 * 4 = 48 cm2 – the area of ​​the larger base.

Truncated pyramid volume:

V = h * (S1 + S2 + √ (S1 * S2)) / 3 = 4 * (12 + 48 + √ (12 * 48)) / 3 = 112 cm3.