Unroll the circle. Reveal the area.

SparkClass

Rearrange equal sectors into a near-parallelogram. No piece stretches, shrinks, or disappears.

CCSS 7.G.B.4
Complete Circle
odd sectors even sectors
Circle sectors rearranged into a near-parallelogram A complete circle made of alternating colored sectors. Moving the rearrange control rotates and translates every unchanged sector into an alternating strip whose base is approximately pi times the radius and whose height is approximately the radius. 16 equal sectors r = 3 units base ≈ ½C = πr ≈ 9.42 units height ≈ r = 3 units
01

Start with one complete circle. The 16 equal sectors alternate in color so you can track every piece.

Radius · r3 units
Circumference · C = 2πr6π ≈ 18.85 units
Exact Area · A = πr²9π ≈ 28.27 units²

Teacher notes · make the limit visible

Keep the distinction explicit: rearranging conserves the sectors’ total area exactly; the parallelogram’s base and height are approximate until we use a limit.

1

Predict before playing

At 0%, ask: “Where will the circumference go when the sectors line up?” Have students trace one arc and predict whether it becomes a long or short edge.

2

Pause and track one piece

Move the progress slider slowly. Choose one color and follow it: each sector only rotates and translates. No area is added, removed, stretched, or scaled.

3

Compare 8 with 32 sectors

Use both proof presets. With 8 sectors, the scalloped edges make the approximation visible. With 32, the long edges look straighter. Ask what improves—and what is still not exact.

4

Name the limiting argument

Half the arcs form each long edge, so base → C/2 = πr. The distance between the long edges → r. Therefore the limiting rectangle has area A = πr · r = πr².

CCSS 7.G.B.4 alignment: this lab supports knowing and informally deriving the formulas for a circle’s area and circumference. Change the radius to verify relationships such as r = 4 giving C = 8π and A = 16π. Decimal values are numerical approximations; expressions in π are exact.