# Hexagonal pyramid

Calculate the surface area of a regular hexagonal pyramid with a base inscribed in a circle with a radius of 8 cm and a height of 20 cm.

Correct result:

S =  674.261 cm2

#### Solution:

$r=8 \ \text{cm} \ \\ a=r=8 \ \text{cm} \ \\ h=20 \ \text{cm} \ \\ S_{1}=3 \cdot \ \sqrt{ 3 }/2 \cdot \ a^2=3 \cdot \ \sqrt{ 3 }/2 \cdot \ 8^2 \doteq 96 \ \sqrt{ 3 } \ \text{cm}^2 \doteq 166.2769 \ \text{cm}^2 \ \\ \ \\ a_{2}=\sqrt{ 3 }/2 \cdot \ a=\sqrt{ 3 }/2 \cdot \ 8 \doteq 4 \ \sqrt{ 3 } \ \text{cm} \doteq 6.9282 \ \text{cm} \ \\ h_{2}=\sqrt{ a_{2}^2+h^2 }=\sqrt{ 6.9282^2+20^2 } \doteq 8 \ \sqrt{ 7 } \ \text{cm} \doteq 21.166 \ \text{cm} \ \\ \ \\ S_{2}=a \cdot \ h_{2}/2=8 \cdot \ 21.166/2 \doteq 32 \ \sqrt{ 7 } \ \text{cm}^2 \doteq 84.664 \ \text{cm}^2 \ \\ \ \\ S=S_{1}+6 \cdot \ S_{2}=166.2769+6 \cdot \ 84.664=674.261 \ \text{cm}^2$

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