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Area of a circle

How can we find the area of a circle when it doesn't split neatly into unit squares? Let's rearrange a circle to make it easier to find its area.

What is the area of a circle?

Unlike squares or rectangles, circles don't have any straight sides. If you draw a circle on graph paper, you'll find that it's hard to get an exact measurement - there are a lot of grid squares that are partly inside the circle, and partly outside. It's not clear how to count them.
Instead, let's start by estimating the area of the circle first. Since calculating the area of a square is easy, we can estimate the area of a circle by comparing it to squares that are smaller and larger than it.
The area of the first square is 72=49 and the area of the second square is 102=100, so we know that the area of the circle must be between 49 and 100. That narrows it down a bit, but we still don't know exactly what the area is. How can we find out?

Building a familiar shape

Let's try cutting up a circle to see if we can rearrange it into a more familiar shape. Look below - see how the four quarters of a circle can be fitted together? The slider controls how many sections we divide the circle into. Slide it to the right to see what happens when we increase that number!
That's right, if we slice up a circle into lots of pieces and shuffle them around, it starts to look like a parallelogram! And we know how to find the area of a parallelogram.
By comparing the circle with the parallelogram, we can clearly see that the height of the parallelogram is the radius, r.
But what's the base?
Hint: Move the slider back to the left and pay close attention to the top and bottom of the wavy parallelogram.
Combined, what part of the circle do the top and bottom of the wavy circle represent?
Choose 1 answer:

That's right, the circumference! As you move the slider back to the right, it should become clear that the top and bottom edges combined are always of length 2πr. But if we just want the base of the parallelogram, we only want half of that length.
What expression represents the length of the base of the parallelogram?

So if the height of our parallelogram is r and the base is πr, what expression represents the area?

Formula for area of a circle

Congratulations, you've just shown that when the radius of a circle is r, then the area of a circle is πrr, which we often write as πr2!
If you ever forget this formula, just think back to the circle that we rearranged into a parallelogram and it will come back right away.

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