Volume and surface area of a right regular pyramid

P2.7.4Volume and surface area of a right regular pyramid

A right regular pyramid with height $$h$$ and a base consisting of a regular $$n$$ -sided polygon of side length $$s$$ has a volume, $V = \frac{1}{3}Ah$ and total surface area, $S = A + \frac{1}{2}nsl$ where $$A$$ is the base area and $$l$$ the slant height which may be calculated from the apothem of the base polygon, $a = \frac{1}{2}s\cot\frac{\pi}{n}$ as $A = \frac{1}{2}nsa$ and $l = \sqrt{h^2 + a^2}$ .

Use these formulae to define a function, pyramid_AV returning $$V$$ and $$S$$ when passed values for $$n$$ , $$s$$ and $$h$$ .

Show solution

The code below is self-explanatory.

import math

def pyramid_AV(n, s, h):
    """ Returns V, S for a right regular polygonal pyramid.

    Returns the volume and total surface area of the right regular pyramid
    of height h with a base of a regular n-gon of side s.

    """

    a = s / 2 / math.tan(math.pi / n)   # apothem
    A = n * s * a / 2                   # base area
    l = math.hypot(h, a)                # slant height
    V = A * h / 3
    S = A + n * s * l / 2
    return V, S

print(pyramid_AV(5, 36.5, 12))

The volume and surface area are reported to be:

(9168.424067738604, 4832.337304213042)