At the base of the pyramid lies a right-angled triangle with legs 12 and 5.

At the base of the pyramid lies a right-angled triangle with legs 12 and 5. All side faces are inclined to the base surface at an angle of 45 °. Find the volume of the pyramid.

We find the area of ​​the base of the pyramid:
S = 1/2 * a * b = 1/2 * 12 * 5 = 30.
By condition, the side faces are inclined at an angle of 45 °, which means that the base of the height of the pyramid coincides with the center of the inscribed circle.
Let’s use another formula for the area of ​​a triangle and find the radius of the inscribed circle:
S = p * r → r = S / p.
p = (a + b + c) / 2.
We find the hypotenuse:
c = √ (144 + 25) = √169 = 13.
p = (12 + 5 + 13) / 2 = 15.
r = S / p = 30/15 = 2.
Consider a right-angled triangle, in which the legs are the height of the pyramid and the radius of the written-off circle, and the hypotenuse is the apothem. The angle between the radius and the apothem is 45 °, which means that the triangle is isosceles:
r = h = 2.
V = 1/3 * S * h = 30 * 2/3 = 20.
Answer: the volume of the pyramid is 20 cubic units.