In a regular triangular prism, the bases of which are 5 6 and 9 cm, the lateral rib has a length of 10 cm

In a regular triangular prism, the bases of which are 5 6 and 9 cm, the lateral rib has a length of 10 cm and forms an angle of 45 degrees with the base plane. Find the volume of the prism.

The prism is simply triangular (an equilateral triangle lies in the regular one at the base and the faces are perpendicular to the base).
Three sides of the triangle are known, let’s use Heron’s formula to find the area.
The half-perimeter of a triangle is:
p = 1/2 * (a + b + c) = 10 (cm).
S = √ (p * (p – a) * (p – b) * (p – c)) = √ (10 * 5 * 4 * 1) = √200 = 10√2 (cm²).
In a right-angled isosceles triangle (the angle between the edge and the base plane is 45 ° by condition), we find the leg, which is the height of the prism:
2h² = 10² = 100
h² = 50
h = √50 = 5√2 (cm).
Find the volume of the prism:
V = S * h = 10√2 * 5√2 = 100 (cm³).
Answer: the volume of the prism is 100 cm³.