11.040. The base of the pyramid is a triangle with side lengths of 6, 5, and $5 \mathrm{~cm}$. The lateral faces of the pyramid form equal dihedral angles with its base, each containing $45^{\circ}$. Determine the volume of the pyramid.
## Solution.

$$
V=\frac{1}{3} S \cdot H, S=\sqrt{p(p-6)(p-5)(p-5)} ; p=\frac{16}{2}=8, \text { therefore } S=12
$$

$S=p \cdot r ;$ hence $r=\frac{12}{8}=\frac{3}{2}$. In $\triangle S O E, \angle S O E=90^{\circ}$ (Fig. 11.35),

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-675.jpg?height=528&width=408&top_left_y=94&top_left_x=186)

Fig. 11.36

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-675.jpg?height=296&width=486&top_left_y=324&top_left_x=713)

Fig. 11.37

$\angle S E O=45^{\circ}$ (by condition) $\Rightarrow E O=S O=\frac{3}{2}$ (since $E O=r$ ), hence $V=\frac{1}{3} \cdot 12 \cdot \frac{3}{2}=6 \mathrm{~cm}^{3}$.

Answer: $6 \mathrm{~cm}^{3}$.