Slice, rearrange, discover
The animation plays automatically: a circle is sliced into 4 → 8 → 16 → 32 wedges and staggered into a row. See it? That row is a near-perfect parallelogram with base πr and height r.
πr² is not a magic spell. Slice a circle into wedges and stagger them and the result is almost a parallelogram with base πr and height r — so the area is πr × r = πr². This animation derives the formula by physical rearrangement.
Aligned with CCSS 7.G.B.4 (formulas for the area and circumference of a circle).
Watch the circle become a parallelogram — that’s the picture behind A = πr².
More slices → the pieces line up into a near-perfect parallelogram (base ≈ πr, height = r).
Cut the circle into 4 equal wedges and lay them in a row, alternating up and down.
Geometry and measurement model
Where Does πr² Come From? is built for students who memorize formulas before seeing the shape decomposition. It gives the page a clear search purpose: learn the model, manipulate it, then continue into the matching grade-level practice.
Where Does πr² Come From? helps when a student can copy a procedure but cannot explain why it works. The demo slows the idea down into a visible model before sending the learner to guided missions.
Learning goals
How to play
Continue with guided practice
Each wedge contributes only half its arc to the bottom edge of the parallelogram (the other half goes to the top edge). So the base = ½ × 2πr = πr.
Every wedge points from the center outward, so its tip is one radius away from its base. After rearranging, that height transfers to the parallelogram.
It's the slice-and-rearrange idea: cut a curved region into infinitely many straight pieces, then sum them. That's the same trick Riemann sums use.
Grades 6–7, aligned with CCSS 7.G.B.4. Foundation for circumference, sector area, and curved-edge composite figures.
