Area of a Circle: A = πr²

The formula

$$A = \pi r^2$$

$$A = \pi(5)^2 = 25\pi \approx 78.5 \text{ square units}$$

Square the radius first, then multiply by $\pi$. The exponent belongs to the $r$ alone — $\pi r^2$ means $\pi \times r^2$, never $(\pi r)^2$.

Why the radius is squared

Area always comes from multiplying two lengths together. That is why a rectangle’s area is $l \times w$ and why the units are squared.

A circle has only one length describing it, so both of the lengths in the product must come from the radius — and $r \times r$ is $r^2$. The $\pi$ is the factor that turns the square into the circle.

You can almost 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 it, and “a bit over three quarters of four” is $\pi$.

$$A_{\text{square}} = 4r^2 \qquad A_{\text{circle}} = \pi r^2 \approx 3.14 r^2$$

Doubling the radius quadruples the area

This is the consequence worth carrying away, and it follows from the squaring.

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

A pizza of twice the diameter is not twice the pizza — it is four times the pizza. Areas grow with the square of the length, which is why a modest increase in size makes a large difference in how much fits inside.

The diameter trap

Given a diameter, halve it first.

$$d = 10 \quad\Rightarrow\quad r = 5 \quad\Rightarrow\quad A = 25\pi$$

Using $10$ where the radius belongs gives $100\pi$ — four times too big, not twice, because the mistaken value is then squared.

That factor of four is the signature of this error. An answer four times the expected size almost always means a diameter went in as a radius.

Working backwards

$$A = \pi r^2 \quad\Rightarrow\quad r = \sqrt{\frac{A}{\pi}}$$

Undo the two operations in reverse order: divide by $\pi$ first, then take the square root. Same reverse-order habit as any two-step equation.

Worked examples

Common mistakes

Practice problems