Circumference of a Circle: C = πd

Quick answer

The circumference is the distance around a circle, found with C = πd or equivalently C = 2πr. Pi is the number you get when you divide any circle's circumference by its diameter — always about 3.14, whatever the size of the circle. A circle of diameter 10 has a circumference of about 31.4.

What you'll learn

Find the circumference from a diameter or a radius
Explain what pi is and why it is the same for every circle
Find a diameter or radius from a known circumference

Perimeter, for a curve

A circle has no sides to add, so perimeter needs a different approach — and its own name. The distance around a circle is its circumference.

A circle with its radius marked

Two measurements describe the size of a circle:

the radius $r$ — centre to edge
the diameter $d$ — right across, through the centre

d = 2r

The formula

C = \pi d \qquad\text{or}\qquad C = 2\pi r

These are the same formula. Since $d = 2r$ , replacing $d$ with $2r$ turns the first into the second. Use whichever matches the number you were given.

A circle of diameter $10$ cm:

C = \pi(10) \approx 31.4 \text{ cm}

What pi actually is

$\pi$ is not a magic constant dropped into a formula. It is a measurement.

Take any circle. Measure its circumference, measure its diameter, and divide:

\pi = \frac{C}{d}

The answer is always about $3.14159$ — for a coin, a wheel, a planet’s orbit.

Circle	Circumference	Diameter	$C \div d$
a coin	$7.54$ cm	$2.4$ cm	$3.14$
a plate	$87.9$ cm	$28$ cm	$3.14$
a wheel	$2.20$ m	$0.7$ m	$3.14$

Why it never changes

Every circle is the same shape, only at different sizes. Doubling a circle doubles the distance around it and doubles the width across it, so the ratio between them is untouched.

That is what makes $\pi$ worth a name: it is the one number every circle has in common. The formula $C = \pi d$ is that definition rearranged.

Working backwards

Knowing the circumference gives the diameter by dividing:

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

A circle with a circumference of $50$ cm has a diameter of $50 \div \pi \approx 15.9$ cm.

Worked examples

Common mistakes

Practice problems

Find the circumference of a circle with diameter $8$ cm. Use $\pi \approx 3.14$ .

Answer

$25.12$ cm

Full solution

$C = 3.14 \times 8 = 25.12$ cm.
Find the circumference of a circle with radius $5$ m. Use $\pi \approx 3.14$ .

Answer

$31.4$ m

Full solution

$C = 2(3.14)(5) = 31.4$ m.
Find the circumference of a circle with diameter $20$ cm, in terms of $\pi$ .

Answer

$20\pi$ cm

Full solution

$C = \pi d = 20\pi$ cm, which is about $62.8$ cm.
A circle has radius $3.5$ m. Find its diameter and circumference.

Answer

Diameter $7$ m, circumference about $22$ m.

Full solution

$d = 2(3.5) = 7$ m, and $C = \pi(7) \approx 22.0$ m.
A circle has a circumference of $31.4$ cm. Find its diameter. Use $\pi \approx 3.14$ .

Hint

Divide.

Answer

$10$ cm

Full solution

$d = \tfrac{31.4}{3.14} = 10$ cm.
A circle has a circumference of $44$ m. Find its radius, to one decimal place.

Answer

About $7.0$ m

Full solution

$r = \tfrac{44}{2\pi} \approx 7.0$ m.
A wheel of diameter $0.5$ m makes $200$ turns. How far does it travel?

Answer

About $314$ m

Full solution

One turn covers $\pi(0.5) \approx 1.571$ m, so $200$ turns cover about $314$ m.
Find the perimeter of a semicircle with radius $6$ cm.

Hint

Do not forget the straight edge.

Answer

About $30.8$ cm

Full solution

Curved part: $\tfrac{1}{2}(2\pi \times 6) = 6\pi \approx 18.85$ cm.

Straight edge (the diameter): $12$ cm.

Total: $18.85 + 12 \approx 30.8$ cm.
A circular track has a radius of $50$ m. A runner covers $4$ laps. How far do they run, to the nearest metre?

Answer

About $1257$ m

Full solution

One lap: $2\pi(50) = 100\pi \approx 314.16$ m.

Four laps: $4 \times 314.16 \approx 1257$ m.
Tom finds the circumference of a circle with radius $7$ cm using $C = \pi d$ and gets $22$ cm. Find his error.

Hint

Which measurement did he put into the formula?

Answer

He used the radius where the formula wants the diameter. The answer is about $44$ cm.

Full solution

$C = \pi d$ needs the diameter, which is $2(7) = 14$ cm:

$C = \pi(14) \approx 44$ cm.

His $22$ cm is exactly half the right answer, which is the signature of this mistake — putting a radius where a diameter belongs always halves the result.

Using $C = 2\pi r$ with the radius he had would have given the same correct $44$ cm.

Frequently asked questions

What is the formula for circumference?

C = πd, where d is the diameter. Since the diameter is twice the radius, C = 2πr says exactly the same thing.

What actually is pi?

The number you get when you divide any circle's circumference by its diameter. It comes out the same for every circle — about 3.14159 — which is why it is worth a name.

Why is pi the same for every circle?

Because all circles are the same shape at different sizes. Scaling a circle up multiplies its circumference and its diameter by the same amount, so their ratio never moves.

Should I use 3.14 or the pi button?

The button, unless the question says otherwise — it carries far more digits. Use 3.14 only when asked, and say which you used.

How do I find the diameter from the circumference?

Divide by pi. If the circumference is 31.4, the diameter is 31.4 divided by π, which is 10.

What to learn next

Area of a Circle: A = πr²How 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.

Formulas on this page

Circumference of a circle

Key terms in this lesson

Pi: Pi is what you get when you divide any circle's circumference by its diameter — always about 3.14159, whatever the size of the circle.
Radius: The radius of a circle is the distance from its centre to its edge. The diameter goes right across through the centre, so it is twice the radius.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.7.G.B.4GeometryKnow the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Frequently asked questions

What is the formula for circumference?

What actually is pi?

Why is pi the same for every circle?

Should I use 3.14 or the pi button?

How do I find the diameter from the circumference?

What to learn next

Formulas on this page

Key terms in this lesson

Standards alignment

More Geometry lessons