Wedges, and a rectangle they approach
A paper circle can be cut into wedges. A wedge is a thin slice with its point at the centre.
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- Cut a circle into 8 equal wedges.
- Lay them side by side. The strip is half the way round long, and the radius wide.
- Look closely. The strip's edges are still curved.
- Same area. Not a rectangle yet.
The rule
A disc is a circle and everything inside it. Call its radius r. Moving the wedges does not change the area. The strip has the same area as the disc. Nothing is lost in the cutting.
Rule: The strip is half the way round long. It is the radius wide.
Half the wedges point up. Half point down. With 8 wedges, 4 point up and 4 point down. Each long edge has 4 arcs, half of all 8. So each edge is half the way round.
| Route | Working | Area |
|---|---|---|
| The strip | (44 ÷ 2) × 7 = 22 × 7 | 154 |
| The circle's area, π × r × r | 22/7 × 7 × 7 | 154 |
Both routes give 154. Half the way round is π × r, and the width is r. A rectangle with these sides has area π × r × r. As the wedges get thin, the strip gets close to that rectangle. The strip keeps the disc's area. So a circle of radius r has area π × r × r.
The strip is not a true rectangle. Each wedge has a curved end, so the long edges are wavy. More wedges make smaller waves. With any number of wedges, some waves remain.