Area of Circles

The two formulas

In the figure below, the segment from the center $O$ out to the circle is the radius $r$. Circumference is a length, so it uses $r$ to the first power: $C = 2\pi r$. Area is two-dimensional, so it uses $r$ squared: $A = \pi r^2$. If you ever blank on which formula has the square, that's the tell — area is measured in square units, so area gets the square.

You'll also see circumference written as $C = \pi d$, where $d$ is the diameter. It's the same formula, because the diameter is just two radii: $d = 2r$.

Radius or diameter? Check before you substitute

The single most common circle mistake is plugging a diameter into a formula that wants a radius. Both formulas above use $r$. If a problem hands you a diameter, cut it in half first — a diameter of $10$ means $r = 5$, and only then does $A = \pi (5)^2 = 25\pi$ come out right. Substituting the $10$ directly gives $100\pi$, which is four times too big.

Read the problem's figure carefully: a segment from the center to the circle is a radius, while a segment all the way across through the center is a diameter. The figure below shows both — the short segment $r$ reaches from the center to the edge, while $d$ runs the full width through the center.

Exact answers vs. decimal answers

An answer like $12\pi$ is exact. An answer like $37.7$ is an approximation, because $\pi$ never terminates. Textbooks and test answer choices usually want the exact form — leave $\pi$ in the answer — unless the problem says to round. A good habit: compute the exact answer in terms of $\pi$ first, then multiply by $3.14159$ only if a decimal is requested.

Keeping $\pi$ symbolic also protects you from rounding errors in multi-step problems, since nothing gets rounded until the very last step.

Worked examples

Try one yourself

Common questions