Surface Area of Pyramids

Date: 05/21/99 at 14:43:28
From: Andy Copley
Subject: Surface Area of Pyramid - Slants?

Dr. Math,

We understand how to find the surface area, but we cannot figure out 
how to determine the slant. The information given is the height and 
the base edge.

Date: 05/21/99 at 16:56:42
From: Doctor Peterson
Subject: Re: Surface Area of Pyramid - Slants?

Hi, Andy. Thanks for your question.

I'll assume you are dealing with a right square pyramid; for other 
bases you can work it out almost the same way.

Here's our pyramid, with base edge a and height h:

                      +
                    //|\
                   ///| \
                 / // |  \
                / //  |   \
              /  / /  |    \
             /  / /   |     \
           /   / /    |      \
          /  s/  /    |h      \
        /    /  /     |        \
       /    /  /      |         \
     /     /   /      |          \
    +-----/---/-------|-----------\
     \   /   /        |            \
      \ /    /        |             \
       +----/---------+              \
        \  /   r                      \
         \ /                           \
          +-----------------------------+
                         a

The height of the slanted face, s, is the hypotenuse of a triangle 
with legs r = a/2 and h. You can use the Pythagorean Theorem to find 
s,

    s = sqrt(h^2 + (a/2)^2)

and then find the area of each face, as/2.

You may notice that if you want the length of the slanted edge for 
some reason, you can use the Pythagorean Theorem again.

If the base were triangular, you would have to find the apothem r by 
different means. I'll let you handle that if you need to.

If you haven't looked at our FAQ on formulas, you'll find the page on 
pyramids here:

  http://mathforum.org/dr.math/faq/formulas/faq.pyramid.html   

This gives formulas for s in the square case.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/