Calculate The Area Of A Circle Simple Explanation

Ever wonder where the formula for calculating the area of a circle came from?    I learned this in public school and thought it was a great way to explain the formula in simple terms.

Lets take an example of a circle with a diameter d.

By definition, the radius is half the length of the diameter of a circle.  The diameter is a straight line that goes through the centre of a circle and each end goes to the circumference of the circle.  The radius is a straight line that goes from the centre of a circle to anywhere on the circumference of the circle.  Written as an equation:

Diameter (d) = 2 x radius (r)

Some guy figured out that if you take the diameter of a circle and wrap it around the circumference, it will go around it 3.14159265359 times (the decimal numbers seem to go on forever, but using 3.14 usually is accurate enough, or you can use the pi button on your calculator).  For example, take a string and cut it the length of the circle's diameter.  Draw the circle to scale on a piece of paper and see how many times the string will wrap around the circumference.  You should find that it goes just over 3.14 times.    See crude sketch below.

Written as an equation:

Circumference (c) = pi (3.14) x diameter (d)

or

Circumference (c) = pi (3.14) x radius (r) x 2

If you cut the circle up like a pie (no I don't think thats where pi comes from) into 16 pieces, it will look like the sketch below:

Now if you take those 16 slices and stack them up, it will look like the sketch below:

The length shown in the vertical is pi x r.  This is because the length is half of the total circumference of a circle (half the circumference is on the left side and half is on the right side) which is 1/2 x pi x d which is the same as 1/2 x pi x 2r (which equals pi x r). 

The length shown in the horizontal is simply the radius of the circle r.

Now that we have converted this circle into a crude rectangle, and imagine that as we make more slices out of the circle and continue until we have infinity (really big number) slices, the circular arcs will now become straight and we will really have a rectangle.

Therefore the formula of a circle is derived from the length x width formula of a rectangle, in this case it is:

Area = length (pi x r) x width (r) = pi x r x r = 

A

=

π

r

2

Note the symbol  π  stands for pi.