Area of a circle

How do we calculate the area of a circle?

We can’t just carve it up into 1cm² because it doesn’t have right angles and it isn’t a portion of a rectangle like a triangle is.

This problem of calculating the area of a circle was attempted by ancient mathematicians by trying to find a square that had the same area as a circle of a certain radius. Thus was called squaring the circle. It didn’t quite work.

Another attempt was made by putting regular polygons inside circles. Like this:

As the number of angles of the polygons increase, the area of the polygon gets closer to the area of the circle. When we have increased the number of angles to twenty then the circle and polygon look like this:

The polygon and circle take up nearly the same amount of area.