The base of the pyramid is a triangle with sides 39cm 17cm 28cm lateral ribs are equal to each other

The base of the pyramid is a triangle with sides 39cm 17cm 28cm lateral ribs are equal to each other 22.9cm find the volume of the pyramid.

Since all side edges are equal, its vertex is projected to the center of the circumscribed circle around the triangle ABC.

Determine the area of ​​the triangle ABC. The semi-perimeter of triangle ABC is equal to:

p = (AB + BC + AC) / 2 = (39 + 28 + 17) / 2 = 42 cm.

According to Heron’s theorem, Sаvs = √42 * (42 – 39) * (42 – 28) * (42 – 17) = √44100 = 210 cm2.

Determine the radius of the circumscribed circle. R = ОА = АВ * ВС * АС / 4 * Sас =

39 * 28 * 17/4 * 210 = 22.1 cm.

In a right-angled triangle AOD, according to the Pythagorean theorem, OD ^ 2 = AD ^ 2 – OA ^ 2 = 524.41 – 488.41 = 36. OD = 6 cm.

Let’s define the volume of the pyramid.

V = Savs * DO / 3 = 210 * 6/3 = 420 cm3.

Answer: The volume of the pyramid is 420 cm3.