Every area formula in the last topic came from cutting a shape up and sliding the pieces into a rectangle. A circle has no straight edges, so that trick seems to fail. It does not fail. It just needs more cuts.
Turning a circle into a rectangle
Cut a paper circle into eight equal sectors and arrange them alternately, one up and the next down, in a row. The result already looks a bit like a parallelogram. With sixteen it looks much more like one, and with thirty two the edges are almost straight.
The height of the rectangle is one radius. The length is made of the curved edges, half pointing up and half pointing down, so it is exactly half the circumference, which is pi r. Area = length times height = pi r times r.
\[ A = \pi r^2 \]
Pi r squared means pi times r times r. Squaring the radius first and then multiplying by pi is the safest order, because it keeps the two operations separate.
Circumference or area?
| Circumference | Area | |
|---|---|---|
| Formula | C = 2πr = πd | A = πr² |
| Units | cm, m | cm², m² |
| It answers | How far round the edge? | How much space inside? |
| Real questions | Wheel distance, fencing a pond, edging a tablecloth | Painting a lid, watering a field, the size of a pizza |
Double the radius and the circumference exactly doubles, but the area becomes four times bigger. This is the rule from Unit 3: lengths scale by k, areas scale by k squared. It explains why a pizza of twice the diameter feeds four people rather than two.
Rings and paths
A ring is a big circle with a smaller one removed. Work out both areas and subtract. Be careful when a question gives the inner radius and the width of the path: the outer radius is the inner radius plus the width.
The formula uses the radius. Questions nearly always give the diameter instead. Halve it first. A diameter of 14 cm put straight into pi r squared gives 616 instead of 154, which is four times too big.
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