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. You 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 along the circle's edge ... 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):
Now we just multply the width by the 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 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 would get to being exactly right.
Area of Circle = π r2