Area of a circle

Square first, then multiply

$\pi r^2$ means $\pi \times r^2$, never $(\pi r)^2$. The exponent belongs to the $r$ alone.

$$r = 5 \quad\Rightarrow\quad A = 25\pi \approx 78.5$$

Why the radius gets squared

Area always comes from multiplying two lengths — that is why a rectangle’s is $l \times w$ and why the units carry a $2$.

A circle is described by one length, so both factors have to come from the radius, and $r \times r$ is $r^2$. The $\pi$ is what converts the square into the circle.

You can nearly see it: a circle of radius $r$ sits inside a square of side $2r$, which has area $4r^2$. The circle fills a bit over three quarters of that — and “a bit over three quarters of four” is $\pi$.

Doubling the radius quadruples the area

Radius	Area
$1$	$\pi$
$2$	$4\pi$
$3$	$9\pi$
$10$	$100\pi$

A pizza of twice the diameter is four times the pizza, not twice. This is the squaring at work, and it is why the larger size is usually better value.

Note that circumference only doubles over the same change — one multiplies a single length, the other multiplies two.

Rearranged

$$r = \sqrt{\frac{A}{\pi}}$$

Undo in reverse order: divide by $\pi$ first, then take the square root.

The signature error

Using a diameter where the radius belongs makes the answer four times too big — not twice, because the mistaken value is then squared. An answer exactly four times the expected size is this mistake.

Rings

Subtract the areas, never the radii:

$$\pi R^2 - \pi r^2 \qquad\text{not}\qquad \pi (R - r)^2$$

For $R = 8$ and $r = 5$: $64\pi - 25\pi = 39\pi$, whereas $(8-5)^2$ would give $9\pi$. See area of a circle.