Surface Area of a Pyramid

Date: 09/01/97 at 19:25:10
From: Daisy Villagomez
Subject: Surface Area of a Pyramid

Hello.  I would just like to know how to get the surface area of a
pyramid. Thanks.

Date: 09/20/97 at 14:21:41
From: Doctor Sonya
Subject: Re: Surface Area of a Pyramid

Hi Daisy!

You didn't say what kind of pyramid, but I'm guessing you want the 
surface area of a regular square pyramid. That's a pyramid with a 
square base and four sides that are all the same. Can you imagine what 
such a pyramid looks like?  

If you have some toothpicks and clay, you can try building one by 
making four toothpicks into a square (held together at the corners 
with clay) and then attaching one toothpick to each corner and 
bringing all four of them together at the top. You can even use 
mini-marshmallows to hold the toothpicks together if you want.  
That's how I build my geometry models. Just in case you don't want to 
build a model, the pyramid has a square base and four equal triangular 
sides. Do you see why the side are triangles?

To get the surface area of the pyramid, you need to find the area of 
the base and the area of each of the sides, and then add them up.  

Whenever you talk about pyramids, there are two things you have to 
know: the length of the sides of the base and the height. These two 
things should be given in your problem.  

The height of a pyramid is how tall it is. If you were to build a 
hollow pyramid, tie a string to the top, and then let it down into the 
middle of the pyramid until it hit the floor, the length of the string 
would be the height. This length is also called an altitude, because 
it is perpendicular to the floor.

Now that we know exactly what we are talking about, let's assign some 
values to our lengths. We'll make them variables becasue you want a 
general formula. Call the length of one side of the base "b" and the 
height of the pyramid "h".

The area of the base is easy to find. What's the area of a square with 
side length b?

Now we need to find the areas of the triangles on the sides. Remember 
that the area of a triangle is:

  (1/2)(base)(height)

The triangles have base b, but what is their height? We can use the 
Pythagorean Theorem to find the height of the triangles.

Picture a triangle inside of the pyramid with one side straight "down" 
through the pyramid from the very top to the center of the base, 
another side from the center of the base to the midpoint of one side 
of the base, and the third side from the midpoint of one side of the 
base to the top of the pyramid.  Drawing a picture (or using more 
marshmallows) might help you visualize this. This third side is the 
height of the triangles we are looking for. Fortunately, we know the 
other two sides: h and b/2.  Think about how we got these. These are 
also two legs of a right triangle.

Remember that the Pythagorean Theorem says that if the two legs of a 
right triangle have length A and B, the the length of the hypotenuse, 
C, can be found with the equation:  A^2 + B^2 = C^2. (A^2 means "A 
squared")

The two legs of our right triangle above are of lengths h and b/2, so 
(length of hypotenuse)^2 = h^2 + (b/2)^2.

If we solve this equation for the length of the hypotenuse, we find 
that it is: SQRT(h^2 + (b/2)^2). So the area of one triangular face 
is: 
 
   (1/2)(base)(height) = (1/2)(b)(SQRT(h^2 + (b/2)^2).

That means the surface area of the entire pyramid is the areas of the 
four triangular faces (remember they're all the same) plus the area of 
the base.

You can use this same technique to find surface areas of pyramids with 
other bases.  Let us know if you need some extra help!

-Doctors Ziggy and Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/