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🍕 Circle Area Lab

Slice a circle like a pizza
and rearrange it — what shape does it become?

Cut a circle into equal slices and alternate flipping them up and down as you line them up, and it gets closer and closer to a rectangle. That rectangle's base is half the circle's circumference (πr), and its height is the radius (r) — so the area is always πr².

Ever wonder where the circle area formula πr² comes from? It's not some big calculation — it comes from a simple idea: cut and rearrange. Divide the circle into equal slices like a pizza, then alternate flipping them up and down. The jagged edge gets smoother and smoother the more finely you cut, getting closer to a rectangle.

That rectangle's width becomes half the original circle's circumference (πr), and its height stays the radius (r). A rectangle's area is width times height, so πr × r = πr² — that's exactly the circle area formula.

The key point: no matter how many pieces you cut it into, the actual area never changes. You only cut it and rearranged where the pieces sit. When there are only a few slices, the rearranged shape's edges are just jagged enough that it doesn't look like a perfect rectangle yet. Try increasing the number of slices with the slider and see for yourself.

Slice pointing up Slice pointing down
Still a circle
Number of slices (m)12
Radius (r)5cm
Area = π × 5²
113.1cm²
Base (half the circumference) ≈ 18.8cm · Height ≈ 6cm