Middle school · Circle geometry
Rearrange a circle to understand its area
Move equal sectors into an alternating strip and see why its limiting dimensions are πr and r.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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Understand what you are seeing
The idea behind the motion.
Cut a circle into equal sectors and rearrange them without stretching. The total area is unchanged. Alternating the sectors produces a strip with curved edges. As the sectors get narrower, those edges become flatter and the strip approaches a rectangle of base half the circumference, πr, and height r. Its limiting area is πr × r.
A relationship to keep
A = πr²; area of one of n sectors = πr²/n
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Cut equal angles
Each sector has angle 360° divided by the number of sectors. Together the sectors still fill exactly one circle.
- STEP 2
Move, never stretch
Move the timeline to translate and rotate each sector into the alternating strip. The rigid pieces keep their individual areas throughout the move.
- STEP 3
Look toward the limit
Increase the number of sectors and compare the flatter strip. For any finite number, its boundaries are still curved. The limiting rectangle has base πr and height r, explaining the circle-area formula.
Your turn to explain
Make a prediction. Test your reasoning.
A circle has radius 3. What are its area and the area of one of 12 equal sectors?
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
The circle has area 9π ≈ 28.274 square units. Each sector has area 9π/12 = 3π/4 ≈ 2.356 square units.
Work through a full lesson
Connect the animation to a worked example and practice questions.