12.360. The slant height of the cone is $l$ and it forms an angle $\alpha$ with the base plane. A sphere is inscribed in this cone, and a regular triangular prism, all of whose edges are equal, is inscribed in the sphere. Find the volume of the prism.
Solution.

The center $K$ of the sphere inscribed in the cone lies on its height $SO, SE$ is the generatrix of the cone, $SE=l, \angle SEO=\alpha$, (Fig. 12.220) $\angle KEO=\frac{1}{2} \angle SEO=\frac{\alpha}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1120.jpg?height=714&width=616&top_left_y=79&top_left_x=108)

Fig. 12.220

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1120.jpg?height=490&width=503&top_left_y=304&top_left_x=758)

Fig. 12.220, a

$KO$ is the radius of the sphere. Let $KO=r$.

In $\triangle SOE\left(\angle SOE=90^{\circ}\right): OE=l \cos \alpha$.

In $\triangle K OE\left(\angle K OE=90^{\circ}\right): r=l \cos \alpha \tan \frac{\alpha}{2}$.

$O_{1}$ and $O_{2}$ are the centers of the bases $A_{1} B_{1} C_{1}$ and $A_{2} B_{2} C_{2}$ of the regular prism inscribed in the sphere (Fig. 12.220, a).

Point $K$ is the midpoint of $O_{1} O_{2}, AK=r$.

Let the edge of the prism be $b$.

Then $O_{1} A_{1}=\frac{b \sqrt{3}}{3}, KO_{1}=\frac{b}{2}$.

In $\triangle A_{1} O_{1} K\left(\angle A_{1} O_{1} K=90^{\circ}\right):$

$A_{1} K^{2}=A_{1} O_{1}^{2}+O_{1} K^{2} ; r^{2}=\frac{b^{2}}{3}+\frac{b^{2}}{4} ; b^{2}=\frac{12 r^{2}}{7} ; b=\frac{2 \sqrt{3} r}{\sqrt{7}}=\frac{2 \sqrt{3} l \cos \alpha \tan \frac{\alpha}{2}}{\sqrt{7}}$.

Volume of the prism:

$V=S_{\triangle A_{1} B_{1} C_{1}} \cdot A_{1} A_{2}=\frac{b^{3} \sqrt{3}}{4}=\frac{24 \sqrt{3} l^{3} \cos ^{3} \alpha \tan^{3} \frac{\alpha}{2}}{7 \sqrt{7}} \cdot \frac{\sqrt{3}}{4}=\frac{18 \sqrt{7}}{49} l^{3} \cos ^{3} \alpha \tan^{3} \frac{\alpha}{2}$.

Answer: $\frac{18 \sqrt{7}}{49} l^{3} \cos ^{3} \alpha \tan^{3} \frac{\alpha}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1121.jpg?height=680&width=454&top_left_y=64&top_left_x=96)

Fig. 12.221

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1121.jpg?height=674&width=697&top_left_y=79&top_left_x=564)

Fig. 12.222