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 =  1521.69 cm3
S =  774.566 cm2

Solution:

h=10 cm r=8 cm n=5  a=2 r sin(π/n)=2 8 sin(3.1416/5)9.4046 cm h1=r2(a/2)2=82(9.4046/2)26.4721 cm  S1=n a h12=5 9.4046 6.47212152.169 cm2  V=S1 h=152.169 10=1521.69 cm3h=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=1521.69 \ \text{cm}^3
S=2 S1+n a h=2 152.169+5 9.4046 10=774.566 cm2S=2 \cdot \ S_{1} + n \cdot \ a \cdot \ h=2 \cdot \ 152.169 + 5 \cdot \ 9.4046 \cdot \ 10=774.566 \ \text{cm}^2



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