Area of a Circle by Cutting into Sectors

Here is a way to find the formula for the area of a circle:

circle 12 sectors

Cut a circle into equal sectors (12 in this example)

Divide just one of the sectors into two equal parts. We now have thirteen sectors – number them 1 to 13:

circle 13 including 2 half slices

Rearrange the 13 sectors like this:

sectors laid out like rectangle

Which resembles a rectangle:

sectors with rectangle on top

What are the (approximate) height and width of the rectangle?

The height is the circle's radius: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.

The width (actually one "bumpy" edge) is half of the curved parts around the circle ... in other words it is about half the circumference of the circle.

We know that:

Circumference = 2 × π × radius

And so the width is about:

Half the Circumference = π × radius

And so we have (approximately):

	radius
π × radius

Now we multply width by height to find the area of the rectangle:

Area = (π × radius) × (radius)

= π × radius²

Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match.

But we can get a better result if we divide the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).

And the more we divide the circle up, the closer we get to being exactly right.

Conclusion

Area of Circle = π r²