Formula

Circumference of a circle

Also written: C = pi d · 2 pi r · distance around a circle

C = \pi d = 2\pi r

The circumference is the distance around a circle: pi times the diameter, or equivalently two pi times the radius.

What each part means

$d$: the diameter, right across through the centre
$r$: the radius, centre to edge — half the diameter
$\pi$: about 3.14159, the same for every circle

When to use it

You need the distance around a circle — a wheel's travel per turn, a track, a rim, the curved part of a semicircle.

Two forms, one formula

Since $d = 2r$ , substituting turns $\pi d$ into $2 \pi r$ . They are the same statement; use whichever matches the measurement you were given.

d = 10 \quad\Rightarrow\quad C = 10\pi \approx 31.4

Pi is a measurement, not a decree

$\pi$ is defined as the answer to a division you could do with string:

\pi = \frac{C}{d}

Measure any circle’s circumference and its diameter, divide, and you get about $3.14159$ — for a coin, a wheel, or a planet’s orbit.

It never changes because every circle is the same shape at a different size. Scaling a circle multiplies its circumference and its diameter by the same factor, so their ratio cannot move.

That makes $C = \pi d$ the definition rearranged rather than a separate fact.

Rearranged

d = \frac{C}{\pi} \qquad r = \frac{C}{2\pi}

Putting a radius into $C = \pi d$ gives exactly half the right answer. Putting a diameter into $C = 2\pi r$ gives exactly double. The two mistakes are mirror images, and the factor tells you which one you made.

A semicircle is not half a perimeter

Half a circle’s curve is $\pi r$ , but its perimeter also includes the straight edge across the cut — the diameter. Cutting the circle open created a new edge:

P_{\text{semicircle}} = \pi r + 2r

See circumference of a circle.

Lessons that teach this

Circumference of a Circle: C = πd
How to find the circumference of a circle from its diameter or radius, what pi actually is, and how to work backwards from a known circumference.
Area of a Circle: A = πr²
How to find the area of a circle from its radius or diameter, why the formula squares the radius, and the mistake of using the diameter instead.