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 =  674.2611 cm²

Solution:

$$\begin{gathered} r=8\text{ cm } \\ a=r=8=8\text{ cm } \\ h=20\text{ cm } \\ {S}_1=3\cdot \sqrt{3}\mathrm{/}2\cdot a^2=3\cdot \sqrt{3}\mathrm{/}2\cdot 8^2=96\sqrt{3}{\text{cm}}^2\doteq 166.2769{\text{cm}}^2 \\ {a}_2=\sqrt{3}\mathrm{/}2\cdot a=\sqrt{3}\mathrm{/}2\cdot 8=4\sqrt{3}\text{ cm}\doteq 6.9282\text{ cm } \\ {h}_2=\sqrt{a_2^2+h^2}=\sqrt{6.9282^2+20^2}=8\sqrt{7}\text{ cm}\doteq 21.166\text{ cm } \\ {S}_2=a\cdot h_2\mathrm{/}2=8\cdot 21.166\mathrm{/}2=32\sqrt{7}{\text{cm}}^2\doteq 84.664{\text{cm}}^2 \\ S=S_1+6\cdot S_2=166.2769+6\cdot 84.664=674.2611{\text{cm}}^2 \end{gathered}$$

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