Why the area of a circle is pi r^2
A circle is sliced into 16 equal sectors, then unfolded and reassembled into a shape resembling a rectangle with base equal to half the circumference (pi r) and height r. As the sector count grows, the bumpy edges smooth into straight lines, showing why the area formula pi r squared holds exactly in the limit of infinitely thin slices. Useful for students meeting circle area proofs for the first time.
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The prompt that made it
A circle cut into 16 sectors is rearranged into a near-rectangle of base pi r (half the circumference) and height r, giving area pi r^2 exactly in the limit of thin sectors.
Make the next one in this series
- Show the same rearrangement using 32 and 64 sectors to see the rectangle converge further
- Animate the analogous slicing proof for the circumference formula using thin triangular wedges
- Demonstrate how this sector method extends to finding the volume of a sphere via disk slices
