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mathcircumference of a circlecircle circumference formularadius and diameterAugust 14, 20266 min read

Circumference of a Circle: How to Use Pi, Radius, and Diameter

By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.

The circumference of a circle means the distance all the way around the circle. You can calculate it with pi times the diameter, or with 2 times pi times the radius, because the diameter is always twice the radius.

From radius or diameter to circumference

If a circle has a diameter of 10 centimetres, multiply 10 by pi. Using 3.14 as an approximation gives a circumference of about 31.4 centimetres. If the same circle is described by its radius, the radius is 5 centimetres. Using 2 times pi times 5 gives the same result, as it should.

The two formulas for circumference of a circle are really the same relationship written in different ways. Diameter equals 2 times radius. Substituting 2 times radius for the diameter turns pi times diameter into 2 times pi times radius. You do not need to memorise two unrelated rules if you understand that connection.

Pi appears because the ratio of a circle's circumference to its diameter is constant for every circle. Measure the distance around a circle and divide by its diameter, and the result approaches pi. This relationship is why circles of different sizes all use the same number in the circumference formula.

How to avoid common circle mistakes

A frequent error is mixing up circumference of a circle with area of a circle. Circumference is a length, so the answer uses ordinary length units such as centimetres or metres. Area measures the space inside the circle, so it uses square units. If your circumference answer ends in square centimetres, that is a sign to check the formula and units.

Another mistake is using a radius value where the formula expects diameter, or using diameter where it expects radius. Circle diagrams sometimes label a line from the centre to the edge, which is a radius, and sometimes a line all the way across through the centre, which is a diameter. Identify which measurement you have before calculating.

You can also estimate before using a calculator. Since pi is a little more than 3, the circumference should be a little more than three times the diameter. A circle with diameter 8 should therefore have a circumference a little above 24. If your calculator gives 6 or 200, the estimate tells you immediately that something went wrong. This quick check makes circumference problems much safer. Reverse problems use the same relationships. If you know the circumference and need the diameter, divide the circumference by pi. If you need the radius, divide by 2 times pi. Keeping the units visible helps here too. A circumference measured in metres produces a diameter and radius in metres. This is a good reminder that the formula describes one geometric relationship that can be rearranged, not a one-way calculation you can only use from radius to circumference. If a problem gives a semicircle or another partial arc, do not automatically use the whole circumference. Work out what fraction of the circle is present, then include any straight edges if the question asks for total perimeter.

The takeaway

The circumference of a circle is the distance around the circle. Use pi times diameter when you know the diameter, or 2 times pi times radius when you know the radius. Keep the answer in length units, remember that diameter is twice the radius, and use the rough rule of a little more than three diameters as a fast sense check.

Practise this

Questions from Geometry and Trigonometry

Reading about something is not the same as being able to recall it. These are real questions from the Geometry and Trigonometry unit in our Math track, answers and explanations included. The unit has 120 in total across 21 steps.

  • Choose all that applyLevel 3

    1. Which of these are volume formulas? Select all that apply.

    • side^3correct
    • length x width x heightcorrect
    • pi x r^2 x hcorrect
    • pi x r^2

    pi x r^2 is the area of a circle; the other three give volumes.

  • Choose all that applyLevel 4

    2. Which statements about circles are true? Select all that apply.

    • diameter = 2 x radiuscorrect
    • circumference = pi x diametercorrect
    • area = pi x r^2correct
    • area = 2 x pi x r

    2 x pi x r is the circumference, not the area; the other three statements are correct.

  • Fact or fibLevel 3

    3. An equilateral triangle has three equal angles of 60 degrees each.

    Answer: True

    All three angles are equal and 180 / 3 = 60, so each is 60 degrees.

See the whole unitPlay it free

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