find the volume of a regular polygon inscribed in a pyramid in python

To find the volume of a regular polygon inscribed in a pyramid, we need to know two things: the height of the pyramid (from the apex to the base), and the area of the base polygon. We can then use the formula V = (1/3)Ah, where A is the area of the base polygon and h is the height of the pyramid.

To find the height of the pyramid, we can use the Pythagorean theorem. Let's say the side length of the base polygon is s, and the apothem (the distance from the center of the polygon to the midpoint of a side) is a. Then the height of the pyramid is given by h = sqrt(s^2 - a^2).

Here's the Python code to compute the volume of a regular polygon inscribed in a pyramid, given the number of sides and the side length:

main.py
import math

def polygon_volume(n, s):
    # compute the apothem
    a = s / (2 * math.tan(math.pi / n))

    # compute the height of the pyramid
    h = math.sqrt(s**2 - a**2)

    # compute the area of the base polygon
    A = 0.25 * n * s**2 * math.tan(math.pi / n)

    # compute the volume of the pyramid
    V = (1/3) * A * h

    return V
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We can then call this function with the number of sides and the side length:

main.py
n = 5  # number of sides
s = 10  # side length
V = polygon_volume(n, s)
print(V)
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This will output the volume of the pyramid with a regular pentagon inscribed in the base.