 # Calculate the volume of a regular pyramid of height h, knowing that this pyramid is based on a convex polygon whose sum of inner angles is n pi and the ratio between the lateral surface and the base area is k. Adrianna Macias 2022-07-23 Answered
Calculate the volume of a regular pyramid of height h
Calculate the volume of a regular pyramid of height h, knowing that this pyramid is based on a convex polygon whose sum of inner angles is nπ and the ratio between the lateral surface and the base area is k.
Idea: Use the formula $\overline{)\frac{lwh}{3}}$ for finding the volume. But at first you have to Bash the problem to find the width and length and then after getting their values you have to plug it into the formula to find the volume.
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Explanation:
The base is a regular polygon with $n+2$ sides. If a is its apothem, then the altitude of every lateral face is ka. From Pythagoras' theorem we then get ${h}^{2}={k}^{2}{a}^{2}-{a}^{2}$ and from this we can find a. Finally, you can find the area of the base as: $A=\left(n+2\right){a}^{2}\mathrm{tan}\frac{\pi }{n+2}.$

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