Area of a Circle

$A = \pi r^2$. Most people meet this as something to memorise, so here is why it is true, in a picture you can cut out of paper. Slice the disc into a great many thin concentric rings and cut each ring open, then lay them in a stack, longest at the bottom. The outermost ring straightens into a strip of length $2\pi r$; the innermost is almost nothing; the ones between shorten evenly. The stack is a triangle of base $2\pi r$ and height $r$, and half base times height gives $\tfrac{1}{2}(2\pi r)(r) = \pi r^2$. The circle's area is not a separate fact from its circumference — it is the circumference integrated outward from the centre, which is also why the derivative of $\pi r^2$ is $2\pi r$.

A worked instance in units you would hold. A circular concrete pad 3.0 m across has $r = 1.5$ m, so $A = \pi(1.5)^2 = 7.07$ m². At 100 mm thick that is 0.71 m³ of concrete, and you would order 0.8 to cover the spill and the screed. Backwards, $r = \sqrt{A/\pi}$ answers the other half of the same job: 12 m² of pad wants a circle 3.91 m in radius, so 7.8 m across.

The squared radius is the part that surprises people, and it answers more real questions than the formula does. Area scales as the square of size, so doubling the diameter quadruples the area. A 2-inch pipe carries four times the cross-section of a 1-inch pipe. A sprinkler that throws twice as far waters four times the lawn. And two 9-inch pizzas (127 in² between them) beat one 12-inch (113 in²), which is the version of this fact most likely to come up over dinner. It also explains why upsizing anything round is rarely a small decision.

Now the mistake, and it is the commonest wrong answer in all of geometry: entering the diameter where the radius belongs. Nobody measures a radius. You measure across the thing, at the widest point if you are lucky, and halve it. Forget the halving and the squaring multiplies your error by four — a tank you thought held 700 L holds 175. Keep one sanity check in your pocket: a circle always fills a bit under 80% of the square that boxes it, because $\pi/4 = 0.785$. So a circle 3 m across sits inside a 9 m² square and must come out around 7 m². If your answer is 28, you used the diameter. The other trap is units — squaring a length squares its unit, so metres in means square metres out, and a radius in centimetres with an area wanted in square metres needs converting first, not at the end.