Hands-on math
A circle's area
A circle's area
hides a little secret
How much fits inside a circle? Watch what happens when we actually cut it apart and rebuild it.
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How much fits inside a circle? Watch what happens when we actually cut it apart and rebuild it.
Four steps turned the circle into a rectangle.
A circle, uncut — like a round pizza fresh out of the oven.
Cut from the center all the way around, like a clock ticking past — lots of identical pieces. Each one has two straight sides and one curved crust.
Lay the pieces tip-up, tip-down, zipping them together like a zipper into one long strip.
The more pieces you cut, the flatter the top and bottom get — and the closer the strip hugs a true rectangle. A rectangle's area is easy: length × width.
The distance all the way around a circle is its circumference — about 3.14 times the distance across. That fixed number is π. So one full trip around is 2πr, and half of it is πr.
That orange crust — half lines the top, half lines the bottom. All together it's one full trip around (2πr), so each long edge is just half: length = πr.
Those two straight sides on every piece are the circle's radius r. Stood upright, they become the rectangle's height: width = r.
Cutting and rebuilding never adds or removes any space. So the circle's area is length × width = πr². The finer you slice, the closer the strip matches that dashed rectangle.