# The base of the pyramid is a right-angled triangle with an acute angle a (alpha).

**The base of the pyramid is a right-angled triangle with an acute angle a (alpha). Two side faces of the pyramid containing the sides of this angle are perpendicular to the base plane, and the third is inclined to it at an angle of b (veta). The height of the pyramid is N. Find the volume of the pyramid.**

Since the faces ACD and ABD are perpendicular to the plane of the base, the edge AD is perpendicular to the sides of AC and AB.

The angle ACB = 90, since the triangle is rectangular, then the angle ACD is the angle of inclination of the BCD face to the base of the pyramid. Angle АСD = β0.

Then in a right-angled triangle АСD tgβ = АД / АС.

AC = AD / tanβ = H / tanβ = H * ctgβ.

In a right-angled triangle ABC tgα = BC / AC.

ВС = АС * tgα = Н * ctgβ * tgα.

Determine the area of the triangle ABC.

Sас = АС * ВС / 2 = Н * ctgβ * Н * ctgβ * tanα / 2 = Н2 * ctg2β * tanα / 2 cm2.

Then V = Saс * АD / 3 = (Н2 * ctg2β * tgα / 2) * Н / 3 = Н3 * ctg2β * tanα / 6 cm3.

Answer: The volume of the pyramid is equal to H3 * ctg2β * tgα / 6 cm3.