The base of the pyramid is a triangle with sides 6, 5 and 5. All lateral ribs are inclined

The base of the pyramid is a triangle with sides 6, 5 and 5. All lateral ribs are inclined to the base at an angle of 45 degrees. Find the volume of the pyramid.

1) Calculate the area of the triangular base.
p = 0.5 * (a + b + c) = 0.5 * (5 + 5 + 6) = 8.
S main = √ (p * (p – a) * (p – b) * (p – c)) = √ (8 * 3 * 3 * 2) = 12.

2) Find the radius of a circle circumscribed around a triangular base:
S = a * b * c / (4 * R);
R = a * b * c / (4 * S) = 5 * 5 * 6 / (4 * 12) = 3.125.

3) Determine the height of the pyramid:
tg 45 ° = h / R;
h = R * tan 45 ° = 3.125 * 1 = 3.125.

4) Now we can find the volume of the pyramid:
V = S main * h / 3 = 12 * 3.125 / 3 = 12.5 cubic meters. units

Answer: the volume of the pyramid is 12.5 cubic meters. units