Informal derivation · Grade 7 geometry
You already use this formula. Here you’ll rebuild it from a circle, a pair of scissors, and one good idea.
Cut the circle into equal slices. Try 4, then 12, then 30. Nothing has been added or thrown away — the slices still hold exactly the area of the original circle.
Drag Rearrange all the way to the right. The slices fan out and interlock, alternating point-up and point-down.
As you use more and more slices, what shape is the rearranged figure approaching?
The top and bottom edges of the parallelogram are the circle’s arcs, split evenly between them. All the arcs together make the whole circumference, 2πr. So how long is just the base?
Every slice still reaches from the centre of the circle out to its edge. What is the height of the parallelogram?
A parallelogram’s area is base × height. The base is πr and the height is r, so the area is πr2 — and the slices always held the circle’s area, so the circle’s area is πr2 too.
No, and this is the interesting part. With 8 slices the edges are visibly bumpy and the answer is only close. With 30 they are almost straight. Nothing ever makes them perfectly straight — but the gap shrinks as far as you like by slicing finer, and that is exactly what it means for the area to be πr2. You have just used a limit without calling it one.