Pentagonal prism

The regular pentagonal prism is 10 cm high. The radius of the circle of the described base is 8 cm. Calculate the volume and surface area of the prism.

Correct result:

V =  1521.6904 cm³
S =  774.5663 cm²

Solution:

$$\begin{gathered} h=10\text{ cm } \\ r=8\text{ cm } \\ n=5 \\ a=2\cdot r\cdot \sin (\pi \mathrm{/}n)=2\cdot 8\cdot \sin (3.1416\mathrm{/}5)\doteq 9.4046\text{ cm } \\ {h}_1=\sqrt{r^2-(a\mathrm{/}2{)}^2}=\sqrt{8^2-(9.4046\mathrm{/}2{)}^2}\doteq 6.4721\text{ cm } \\ {S}_1=n\cdot {\displaystyle \frac{a\cdot h_1}{2}}=5\cdot {\displaystyle \frac{9.4046\cdot 6.4721}{2}}\doteq 152.169{\text{cm}}^2 \\ V=S_1\cdot h=152.169\cdot 10=1521.6904{\text{cm}}^3 \end{gathered}$$

$S=2\cdot S_1+n\cdot a\cdot h=2\cdot 152.169+5\cdot 9.4046\cdot 10=774.5663{\text{cm}}^2$

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