At the base of the pyramid lies a right-angled triangle with legs 6 and 8 cm. The side faces are equally inclined to the base

At the base of the pyramid lies a right-angled triangle with legs 6 and 8 cm. The side faces are equally inclined to the base at an angle of 45 degrees. Find the volume of the pyramid.

Determine the area of ​​the base of the pyramid.

Sbn = AB * BC / 2 = 8 * 6/2 = 24 cm2.

By the Pythagorean theorem, we determine the length of the hypotenuse AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 = 64 + 36 = 100.

AC = 10 cm.

Since all side faces are inclined to the base at the same angle, the top of the pyramid is projected to the center of the circle inscribed in the base triangle.

Let’s define the radius of this circle.

Since the ABC triangle is rectangular, OH = r = (AB + BC – AB) / 2 = (8 + 6 – 10) / 2 = 2 cm.

The triangle DOH is rectangular and isosceles, since the angle O = 90, the angle H = 45.

Then ОD = ОН = 2 cm.

Then Vpir = Sbn * OD / 3 = 24 * 2/3 = 16 cm3.

Answer: The volume of the pyramid is 16 cm3.