What it does
Why is the area of a circle πr²? Cut a circle into equal slices like a pizza and lay them in a row, one pointing up, the next pointing down. The row looks a bit like a parallelogram:
- its height is about the radius, r;
- its width is half the slices’ curved edges, which is half the circumference: ½ × 2πr = πr.
So the area is about πr × r = πr². With more and thinner slices, the bumps along the top and bottom flatten out, the slanted ends stand up, and the row becomes a true rectangle. This page lets you watch that happen with 4, 8, 16 or up to 200 slices, and shows the numbers getting closer to πr².
It follows the Class 8 maths chapter Area (area of a circle) and helps with circle areas in Class 10. Why sliding a shape keeps its area is in shear and area, and the area of one slice (a sector) is in circle sectors.
How to use it
- Set the number of slices n (always even, so half point up and half point down).
- Set the radius r (in cm, or any unit).
- Move Unroll to slide the slices between the circle and the row, or press Play.
- Read the width, height and width × height of the row and compare them with πr².
A worked example
The page starts with r = 7 cm and 8 slices.
- Each slice’s straight edge (chord) is 2 × 7 × sin(22.5°) ≈ 5.358 cm. Four slices point up, so the row is about 4 × 5.358 = 21.43 cm wide.
- The height from tip to chord is 7 × cos(22.5°) ≈ 6.467 cm.
- Width × height ≈ 138.6 cm², which is 90.03% of πr² = π × 49 ≈ 153.94 cm².
With 64 slices the row is 21.98 cm wide (π × 7 = 21.99) and 6.99 cm tall, and width × height is 99.84% of πr². With 200 slices it is 99.98%.
Tips
- The curved top of the row is always exactly half the circle’s edge, πr. Only the straight-line measurement here falls a little short with few slices.
- Area grows with the square of the radius: double r and the area is four times as big.
- With 22/7 for π, a radius that is a multiple of 7 gives tidy answers: r = 7 gives 154 cm².
Check yourself
Four fresh questions: an area from the radius, an area from the diameter, the width of the unrolled row, and what happens when the radius grows. Nothing is saved.
FAQ
Is this a proof? It is the idea behind one. To make it a proof you show the error shrinks to zero as the slices get thinner, which is a limit, met properly in Class 11 and 12.
Is this the NCERT book? No. It covers the same topic with fibub’s own pictures and questions. fibub is not part of NCERT.
Where the numbers come from
- The slicing argument for the area of a circle and the inscribed-polygon area (n/2)·r²·sin(2π/n) are standard geometry. Pictures and questions written for fibub.
- Checked: width × height of the row equals the inscribed regular n-gon area (n/2)·r²·sin(2π/n) and tends to πr² (within 0.02% at n = 200), in Python and in tests/sci-tools/circle-slices.mjs.
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Last updated 2026-11-09.