Area and circumference of a circle

The circle is the one figure in this branch whose area cannot be assembled out of rectangles — and that is why its formula contains a number that cannot be written down exactly: $\pi$.

Radius, diameter, centre

The radius $r$ is the segment from the centre to the edge. The diameter $d$ runs edge to edge through the centre, so it is exactly twice as long:

The names are worth keeping apart too: a circle is the line itself, a disc is that line together with its interior. Hence "the area of a disc", but "the length of a circle".

The number π

Divide any circle's circumference by its diameter — you always get the same number. A plate, a coin, a bicycle wheel: the answer is identical.

$\pi$ is irrational: its decimal expansion never ends and never repeats. In school arithmetic we take $\pi \approx 3.14$.

Circumference

Since $\pi$ is the circumference divided by the diameter, the circumference is the diameter multiplied by $\pi$:

A circumference is a length, so it is measured in centimetres and metres — never in square units. If your data is in inches or millimetres, convert it first with our length converter.

Area

The square applies to the radius alone, not to the product $\pi r$. The order is unambiguous: first $r^2$, then multiply by $\pi$.

Watch out for the diameter

This is the costliest slip in the topic. If a problem gives you the diameter, halve it first.

Rounding

Because $\pi$ is irrational, the answer is almost never a "nice" number. Two spellings are correct:

• exact, with $\pi$ left in: $25\pi\,\text{cm}^2$;
• approximate, once $\pi \approx 3.14$ is substituted: $78.5\,\text{cm}^2$.

The exercises below ask for the approximate value, rounded to two decimal places.

Exercises

Type the number alone — the unit is stated in the question — and round to two decimal places. An answer worked out with $\pi \approx 3.14$ is accepted too.

Common mistakes

• The diameter substituted for the radius — the area then comes out four times too big.
• $(\pi r)^2$ instead of $\pi r^2$ — only the radius is squared.
• Mixing the formulas up — $2\pi r$ is the circumference, $\pi r^2$ is the area; the area is the one with the square.
• An area unit without the square — $78.5\,\text{cm}^2$, not $78.5\,\text{cm}$.
• Rounding mid-calculation — compute first, round last.
• Treating π as 3 — $\pi \approx 3.14$; "about three" is fine for an estimate, not for an answer.