Where does the formula come from?
Archimedes derived the formula for the area of a circle by viewing it as the area of a regular polygon with infinitely many sides.
If you want to explore the area of a circle formula in a hands-on way, choose one of the following
- Print out this fraction circle worksheet.
- Draw your own circle and split it up in to equal-sized pieces.
- Open this applet and manipulate the circle virtually.
Cut up a circle into 6, 8, 10, or 12 equal-sized pieces and then rearrange the pieces side by side like this…
When the pieces are arranged like this, the circumference of the circle is split into two equal lengths shown by the red, curvy lines.
Notice that the rearranged pieces create the rough shape of a parallelogram.
It is not a perfect parallelogram because the red circumference lines are curvy instead of straight. However, if you cut the circle into smaller pieces, the rearranged shape would look even more like a parallelogram.
If you could somehow cut the circle into an infinite number of pieces, then the rearranged shape would be a perfect parallelogram.
You can play around with this idea by exploring this applet that allows you to cut up a circle into 200 pieces!
The height of the parallelogram is the radius of the circle (r).
The base of the parallelogram is approximately half of the circumference of the circle. It is not exact because the red lines are curved, but it is pretty close.
If you cut the circle up into smaller pieces, the red curvy line will look more and more like a straight line and the parallelogram will look more and more like a rectangle.
The formula for the circumference of the circle is \(2\pi r\). and half of that is \(\pi r\). This is the base of the parallelogram that is made from the cut-up pieces of the circle.
To find the area of the parallelogram, multiply the base of the parallelogram (\(\pi r\)) by the height (r).
The area of the parallelogram (\(\pi r\)\(r\) or \(\pi r^{2}\)) is the same area as the circle because the pieces of the circle made the parallelogram. So, that is why the formula for the area of a circle is \(\pi r^{2}\).
