Area of a Circle

Let's take a look at how the formula for the area of a circle makes sense.

First, check out what happens if we cut a circle into sectors and rearrange them. Notice how if we make more sectors, our arrangement starts to look more and more like a rectangle.

If we make 98 sectors, now it really, really looks like a rectangle

Now, what will be the base and height of the rectangle?

  • Height: we can see that as we make more and more sectors, the radius of a sector becomes the same thing as the height of the rectangle. Therefore,

    • Height = r

  • Base: we can see that the arcs of the green sectors form a line on top of the rectangle, while the arcs of the yellow sectors form a line on the bottom of the rectangle. Since half of the sectors are yellow, that means half the the arcs are yellow. If half of the arcs are yellow, that means half of the circumference is yellow. Soooo, since the bottom half of the rectangle is the yellow arcs, that means it's equal to half the circumference of the circle

    • Base = (1/2)C = (1/2)2πr

The area of a rectangle is base x height. Let's see what we get.

A = b x h

Substituting

A = r x (1/2)2πr

Simplifying and rearranging

A = πr2

Interesting. And since the rectangle is just a rearrangement of the circle, that means the area of the circle is also A = πr2. And of course, this is the famous area of a circle formula

Summary

  • We can cut a circle into equal sectors and rearrange them into a rectangle shape

  • As we cut smaller sectors, the height of the rectangle becomes equal to the radius of the sectors

  • As we cut smaller sectors, the arcs of the sectors become equal to the top and bottom sides of the rectangle

    • Therefore the length of the rectangle is half the of the arcs, which is half of the circumference

  • The area of the rectangle is then

    • A = b x h

    • A = r x (1/2)2πr

    • A = πr2