Circle Formulas: Area and Perimeter

Date: 03/19/2003 at 16:52:32
From: Leo
Subject: Math formulas

I get confused with the different formulas, such as Pi r squared or 
Pi d (Pi times radius squared or Pi times diameter). I don't remember 
which formula goes to which problem.

Date: 03/21/2003 at 13:45:56
From: Doctor Dotty
Subject: Re: Math formulas

Hi Leo,

Thanks for the question!

The best way is to actually understand what each equation is actually 
doing.

CIRCUMFERENCE:

It was discovered that the ratio between the diameter of a circle and 
the circumference (distance around the outside) was constant. What 
this means is that you can multiply the diameter of a circle by a 
special number that is just over 3 to find the distance around the 
outside. The amazing thing is that this number is the same, no matter 
how big the circle is! This number was given the name 'Pi' and is 
about 3.14159.  So 

  circumference = Pi * diameter

The diameter is twice the radius (as the radius measures from the 
centre to the outside of a circle, and the diameter measures straight 
across the circle through the centre) so this equation can be written 
as:

   circumference = Pi * 2 * radius

Now, can you see where that all comes from? It is just multiplying a 
length (the radius) by a number to get a bigger length (the 
circumference)

AREA:

Here is a circle, with a sector (the shape of a slice of pizza) 
drawn in.


                 *   *   *
             *              . *
          *                .   . *
        *                 .   .     *
       *                 .  .        *
      *                 . . r        *
      *                ..            *
      *               .              *
      *                              *
      *                              *
       *                            *
        *                          *
          *                      *
             *                *
                 *   *   *

Let's call the angle between the two radii x, and the curved length 
at the end b.

Imagine that the angle x gets extremely small. As it gets smaller and 
smaller, b gets closer and closer to a straight line. By the time the 
angle is nearly 0, b is very close to a straight line.

So that makes the sector into a triangle. Now imagine splitting the 
circle into lots of sectors like the first, and putting the resulting 
triangles into a row. Here's a simplified drawing: 

    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 
   |\     |\     |\     |\     |\     |   
   | \    | \    | \    | \    | \    |     
   |  \   |  \   |  \   |  \   |  \   |      
   |   \  |   \  |   \  |   \  |   \  |    
   |    \ |    \ |    \ |    \ |    \ |     
   |_ _ _\|_ _ _\|_ _ _\|_ _ _\| _ _ \|


This gives a rectangle with the same area as the circle. Now, looking 
at the original sector in the circle we can see that the sides of this 
rectangle are equal to the radius of the circle. The side along the 
top is made from all the little bits of the circumference. When all 
the sectors are in here, then the length of the top side of the 
rectangle is half of the circumference. Let's call the circumference 
c, and call the area 'a'. We know that the area of a rectangle (and 
therefore the area of the circle) is: 

    area = height * width 

So:

     a = r * c/2 

But we know that c = Pi * d    [from the circumference bit above]

So:
    a = r * (Pi * 2 * r)/2

    a = r * 2(Pi * r)/2

    a = r * (Pi * r)

    a = r * Pi * r

    a = Pi * r^2  [where ^2 means squared]

Can you see where that comes from now? Another way to remember that 
this equation regards the area is that the radius is squared. The 
radius could be measured in metres, so the result will be measured 
in metres squared. The only thing you measure in metres squared is 
area.

Does that make sense? 

If I can help any more with this problem or any other, please write 
back!

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