A circle's area is found
using the formula:
But where does this formula come
from?
Let's find out ... |

What we're going to do is break up
a circle into little pieces, and then reassemble it into a shape that we
know the area formula for ... the rectangle.
Maybe you're wondering how on earth you
can rearrange pieces of a circle to make a rectangle! Well, just
watch ... it's easy!
We'll start with the circle that we want to break up: Let's break the circle into eighths instead:
these pieces into a rectangular shape:
look like a rectangle
... but we're not there yet! The next step is to go back and try splitting
the circle into sixteenths. Here are the pieces:
A = L x W
... but this shape does not have straight sides, so the formula
wouldn't be very accurate.
Let's go one step further, and break up the circle
into a still not perfectly straight
... they are definitely a little bumpy.
Can you visualize what would happen if we kept going? If we continued to break the circle up into tinier and tinier pieces? Eventually, the bumps would become so small that we couldn't see them, and the top and bottom of the shape would appear perfectly straight. This is what we would see: 'How long are the length
and width of our rectangle made from circle parts?'
Let's go back to an earlier picture, so you can see the circle parts more clearly: top of the 'rectangle',
and the other half of the circle, also length ,
goes on the bottom
So we know the length is
and the width is Now we can find the area of the shape, using the rectangle formula: ... and there we have the formula for the area of the circle we started with! |