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.

Result

V =  1521.69 cm³
S =  774.566 cm²

Solution:

$h=10 \ \text{cm} \ \\ r=8 \ \text{cm} \ \\ n=5 \ \\ \ \\ a=2 \cdot \ r \cdot \ \sin(\pi/n)=2 \cdot \ 8 \cdot \ \sin(3.1416/5) \doteq 9.4046 \ \text{cm} \ \\ h_{1}=\sqrt{ r^2-(a/2)^2 }=\sqrt{ 8^2-(9.4046/2)^2 } \doteq 6.4721 \ \text{cm} \ \\ \ \\ S_{1}=n \cdot \ \dfrac{ a \cdot \ h_{1} }{ 2 }=5 \cdot \ \dfrac{ 9.4046 \cdot \ 6.4721 }{ 2 } \doteq 152.169 \ \text{cm}^2 \ \\ \ \\ V=S_{1} \cdot \ h=152.169 \cdot \ 10 \doteq 1521.6904 \doteq 1521.69 \ \text{cm}^3$

$S=2 \cdot \ S_{1} + n \cdot \ a \cdot \ h=2 \cdot \ 152.169 + 5 \cdot \ 9.4046 \cdot \ 10 \doteq 774.5663 \doteq 774.566 \ \text{cm}^2$

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