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.

Result

S =  674.261 cm2

Solution:

r=8 cm a=r=8 cm h=20 cm S1=3⋅ 3/2⋅ a2=3⋅ 3/2⋅ 82≐96 3 cm2≐166.2769 cm2  a2=3/2⋅ a=3/2⋅ 8≐4 3 cm≐6.9282 cm h2=a22+h2=6.92822+202≐8 7 cm≐21.166 cm  S2=a⋅ h2/2=8⋅ 21.166/2≐32 7 cm2≐84.664 cm2  S=S1+6⋅ S2=166.2769+6⋅ 84.664≐674.2611≐674.261 cm2r=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 \doteq 674.2611 \doteq 674.261 \ \text{cm}^2

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