Circumference of a Circle

The distance all the way around a circle equals pi times the distance across it.

C = π d = 2π r

Circumference is the length of the outer edge. The diameter d cuts straight across.

What each symbol means

Circumference is a fancy word for a simple idea: the distance around the outside of a circle. If you took a piece of string and laid it exactly around a circle, then straightened it out, the length of that string is the circumference.

You can find it two ways, and they give the same answer. If you know the diameter, use C = π d. If you know the radius, use C = 2π r. They match because the diameter is just two radii: d = 2r.

Where pi comes from

Here is the surprising part. If you measure the distance around any circle and divide it by the distance across, you always get the same number: about 3.14. Big circle, small circle, it does not matter. That fixed number is pi. So circumference is simply pi times the diameter.

One formula, two forms: Use C = π d when a problem gives the diameter. Use C = 2π r when it gives the radius. Pick whichever matches the number you were handed.

A worked example

A bike wheel has a radius of 30 cm. How far does it travel in one turn?

  1. One full turn covers exactly one circumference.
  2. Use C = 2π r because we know the radius.
  3. Put in the numbers: C = 2 × 3.14 × 30.
  4. Multiply: 2 × 3.14 = 6.28, then 6.28 × 30 = 188.4.
  5. The wheel travels about 188.4 cm in one turn.

A second example, from the diameter

A round pond is 14 m across. What is the distance around it?

  1. The 14 m is the diameter, so use C = π d.
  2. With π ≈ 227: C = 227 × 14.
  3. Simplify: 14 ÷ 7 = 2, so C = 22 × 2 = 44.
  4. The distance around the pond is about 44 m.

Why pi shows up

  1. Take any circle and carefully measure two things: the distance around it (circumference) and the distance straight across it (diameter).
  2. Divide the distance around by the distance across. You get about 3.14.
  3. Try it on a bigger or smaller circle. You still get about 3.14. This ratio never changes, and we name it pi.
  4. So "circumference divided by diameter equals pi" rearranges to "circumference equals pi times diameter", which is C = π d.

Practice questions

Question 1

Find the circumference of a circle with diameter 10 cm. Use π ≈ 3.14.

  1. We are given the diameter, so use the form C = π d.
  2. Put in the numbers: C = 3.14 × 10.
  3. Multiply to get C = 31.4.
  4. The circumference is 31.4 cm.

Answer: 31.4 cm

Question 2

Find the circumference of a circle with radius 5 m. Use π ≈ 3.14.

  1. This time we are given the radius, so use the form C = 2π r.
  2. Put in the numbers: C = 2 × 3.14 × 5.
  3. Multiply step by step: 2 × 3.14 = 6.28, then 6.28 × 5 = 31.4.
  4. The circumference is 31.4 m.

Answer: 31.4 m

Question 3

A circle has diameter 7 cm. Find its circumference using π ≈ 227.

  1. We are given the diameter, so use C = π d.
  2. Put in the numbers with π ≈ 227: C = 227 × 7.
  3. The 7 on top cancels the 7 on the bottom, leaving C = 22.
  4. The circumference is 22 cm.

Answer: 22 cm

Question 4

A wheel has radius 21 cm. How far does it roll in one turn? Use π ≈ 227.

  1. One full turn rolls out exactly one circumference, and we know the radius, so use C = 2π r.
  2. Put in the numbers with π ≈ 227: C = 2 × 227 × 21.
  3. Simplify the fraction first: 21 ÷ 7 = 3, so this becomes 2 × 22 × 3.
  4. Multiply: 2 × 22 = 44, then 44 × 3 = 132.
  5. The wheel rolls 132 cm in one turn.

Answer: 132 cm

Question 5

The circumference of a circle is C = π d. If the diameter doubles, what happens to the circumference?

  1. Look at the formula C = π d: the diameter is multiplied by the fixed number π.
  2. When one quantity is just a fixed number times another, they change together in step.
  3. So replacing d with 2d gives π × 2d = 2 × (π d), which is twice the old circumference.
  4. The circumference also doubles.

Answer: It also doubles.