# Area of a Circle by Cutting into Sectors

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

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:

Rearrange the 13 sectors like this:

Which resembles a rectangle:

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 just multply the width by the height to find the area of the rectangle:

Area = (π × radius) × (radius)

= π × radius^{2}

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

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

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

## Conclusion

Area of Circle = π r^{2}