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 $k$.

Idea: Use the formula $\boxed{\frac{lwh}{3}}$ 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.

  • 1
    $\begingroup$ "...and the relationship between the lateral surface and the base area is k." means what exactly? What is the question too - do you know the answer and are giving us a hint?? $\endgroup$
    – Paul
    Dec 5, 2019 at 12:47
  • $\begingroup$ @Paul The question is to calculate the volume of the pyramid. This is not a tip, it is my idea to solve the problem, what stops me is algebra (solving the expressions). $\endgroup$
    – Sullo
    Dec 5, 2019 at 12:49
  • $\begingroup$ The problem seems to describe a pyramid with a regular $(n+2)$-gon shaped base. $\endgroup$
    – random
    Dec 5, 2019 at 14:30

1 Answer 1



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 $h^2=k^2a^2-a^2$ and from this we can find $a$. Finally, you can find the area of the base as: $$A=(n+2)a^2\tan{\pi\over n+2}.$$


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