Area of a Circle = πr² — Unrolling Sectors into a Rectangle
Slice a circle like a pizza and lay the sectors alternately: it approaches a rectangle of base πr and height r — area πr².
Do not write an equation yet. Shake the figure first.
Increase the pizza slices to 64
Alternating arcs flatten toward a rectangle whose base is πr and height is r.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
Slice a circle of radius r into 8 sectors. Its area?
Return to the exact question
Show that a circle of radius r has area πr² by cutting it into sectors and rearranging them.
Why this approach works
Cut the circle into thin sectors and lay them alternately up and down: a jagged strip forms whose base is half the circumference, πr, and whose height is the radius r. As the number of sectors grows without bound, the jaggedness vanishes and the strip becomes a perfect rectangle, πr by r — so the circle’s area is πr × r = πr². Turning curves into straight pieces is the seed of integration.
Proof
Rearranging the sectors alternately gives a strip of base πr (half the circumference) and height r. In the limit of infinitely many sectors it becomes an exact rectangle of area πr × r = πr².
Try it yourself
Imagine 8, then 16, then 64 sectors: the wavy top and bottom flatten, converging to a rectangle.
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