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:
Now split the circle into quarters:
Then reassemble them to try to make a rectangle:
Not exactly a rectangle, is it? But we're not done yet!
Let's break the circle into eighths instead:
... and arrange these pieces into a rectangular shape:
This is certainly starting to 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:
This time when we put them together, they are much closer to looking like a rectangle! See what you think:
The goal is to make a shape that is as close to a rectangle as possible, so that we can find its area using the rectangle formula 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 whole bunch of little pieces. When we rearrange all the pieces, the shape would look something like this:
This is very close to a perfect rectangle! But you can see that the top and bottom are 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:
A perfect rectangle! Now all we have to do is find its area, using the formula A = L x W
The next question is, '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:
The original circle's outside length was the distance around, or the circumference of the circle:
Half of this distance around, , goes on the top of the 'rectangle', and the other half of the circle, also length , goes on the bottom
In other words, all of the red and blue pieces add up to , the circumference.
The sides are just the radius of each of the pieces, or the radius of the circle, r.
So we know the length is and the width is r
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!