11.080. On the bases of a cylinder with a square axial section, two cones are constructed with vertices at the midpoint of the axis (of the cylinder). Find the sum of the total surface areas and the sum of the volumes of the cones, if the height of the cylinder is $2 a$.
## Solution.

By construction $H=2 R$ (Fig. 11.72). These cones are identical. The area of the total surface of one cone: $S_{1}=S_{\text {lat }}+S_{\text {base }}=\pi R l+\pi R^{2}$. From $\triangle A B C \quad 2 l=2 \sqrt{2} R$. Hence $l=\sqrt{2} R$. Since $2 R=2 a$, then $S_{1}=\pi a^{2} \sqrt{2}+\pi a^{2}$. And the sum of the total surfaces $S=2 \pi a^{2}(\sqrt{2}+1)$. The volume of one cone $V_{1}=\frac{1}{3} \pi R^{2} H=\frac{1}{3} \pi a^{3}$. And the sum of the volumes of the cones $V=\frac{2}{3} \pi a^{3}$

Answer: $2 \pi a^{2}(\sqrt{2}+1) ; \frac{2}{3} \pi a^{3}$