MISSION · Geometry
Circumference and Area of a Circle
Where pi comes from, and why one formula squares the radius
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Every circle, no matter its size, is a little more than three times as far around as it is across — and that stubborn ratio is \(\pi\).
Pi is a measurement, not a magic number
Wrap a string around a jar lid, then lay the string next to the lid’s width. The string is about 3.14 lid-widths long. Do it with a coin, a plate, a bicycle wheel, a crater on the moon. Same answer, every time.
\[\pi = \frac{\text{circumference}}{\text{diameter}} \approx 3.14159\]Rearranged, that’s the first formula:
\[C = \pi d = 2\pi r\]Nobody invented \(\pi\). It was measured, and then proved to go on forever without repeating. You will almost always use 3.14, or leave the symbol in place.
Why area squares the radius
\[A = \pi r^2\]Here’s the picture that makes it believable. Cut a circle into many thin wedges, like a pizza, and alternate them point-up, point-down. The wedges lock together into something very close to a rectangle. That rectangle is half the circumference long (\(\pi r\)) and one radius tall (\(r\)), so its area is \(\pi r \times r = \pi r^2\). The thinner you cut the wedges, the more exactly it becomes a rectangle.
The radius is doing all the work
Double the radius and the circumference doubles, because \(r\) appears once. But the area quadruples, because \(r\) appears twice. A 16-inch pizza is not a bit more food than an 8-inch pizza — it is four times the food.
| Radius | Circumference | Area |
|---|---|---|
| 2 | 12.57 | 12.57 |
| 4 | 25.13 | 50.27 |
| 8 | 50.27 | 201.06 |
Those two equal numbers in the first row are a coincidence of \(r = 2\), and a good reminder to keep track of units: one is a length, the other is squared.
Common mix-ups
- Using the diameter where the formula wants the radius. \(A = \pi d^2\) gives four times too much. Halve the diameter first, every time.
- Squaring \(\pi r\) instead of \(r\). In \(\pi r^2\), only the radius is squared.
- Swapping the two formulas. The one with the square is the one about the space inside, because area is always measured in square units.
- Rounding \(\pi\) to 3 and calling it exact. It’s fine for a rough check, not for an answer.
Check yourself
A circular garden has a radius of 5 metres. How much fence goes around it, and how much soil covers it?
Fence: \(C = 2\pi(5) \approx 31.4\) metres. Soil: \(A = \pi(5)^2 \approx 78.5\) square metres.
LEVEL 3 · REFLECT
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