12.254. The base of the pyramid is a rhombus with an acute angle $\alpha$. All lateral faces form the same angle $\beta$ with the base plane. Find the radius of the sphere inscribed in the pyramid if the volume of the pyramid is $V$.
## Solution.

Let the rhombus $ABCD$ be the base of the pyramid $EABCD$, $\angle BCD = \alpha$, $0^{\circ} < \alpha < 90^{\circ}$, and $EO$ be its height (Fig. 12.120). Since all lateral faces of the pyramid form the same angle $\beta$ with the base plane, point $O$ is the center of the circle inscribed in the rhombus $ABCD$, which is the point of intersection of its diagonals, and the points of the height $EO$ are equidistant from the lateral faces of the pyramid.

In the plane $CED$, drop a perpendicular $EF$ to $CD$. $OF$ is the projection of $EF$ onto the base plane. Then $OF \perp CD$ and $\angle EFO$ is the angle of inclination of the face $CED$ to the base plane, $\angle EFO = \beta$.

If $O_1$ is the point of intersection of the bisector of $\angle EFO$ with the height of the pyramid, then $O_1$ is the center of the sphere inscribed in the pyramid, and $O_1O$ is its radius.

Let $CD = a$, then from

$$
\triangle COD (\angle COD = 90^{\circ}): CO = CD \cos \angle OCD = a \cos \frac{\alpha}{2}
$$

In $\triangle OFC (\angle OFC = 90^{\circ})$:

$OF = OC \sin \angle OCD = a \cos \frac{\alpha}{2} \sin \frac{\alpha}{2} = \frac{1}{2} a \sin \alpha$.

In $\triangle EOF (\angle EOF = 90^{\circ})$:

$EO = OF \tan \angle EFO = \frac{1}{2} a \sin \alpha \tan \beta$.

The area of the base of the pyramid $S = CD^2 \sin \angle BCD = a^2 \sin \alpha$.

The volume of the pyramid $V = \frac{1}{3} S \cdot EO = \frac{1}{6} a^3 \sin^2 \alpha \tan \beta \Rightarrow a = \frac{\sqrt[3]{6 V \sin \alpha \cot \beta}}{\sin \alpha}$.

In $\triangle O_1OF (\angle O_1OF = 90^{\circ})$:

$OO_1 = OF \tan \angle OFF_1 = \frac{1}{2} a \sin \alpha \tan \frac{\beta}{2} = \frac{1}{2} \tan \frac{\beta}{2} \sqrt[3]{6 V \sin \alpha \cot \beta}$.

Answer: $\frac{1}{2} \tan \frac{\beta}{2} \sqrt[3]{6 V \sin \alpha \cot \beta}$.