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Circumference of a circle

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A circle with a dashed diameter across its middle labelled 3.6 m3.6 m
Diameter of the trampoline
\(\displaystyle 3.6\text{ m}\)

A circular trampoline has diameter 3.6 m and a safety pad runs around its edge. What is the length of the pad, to 2 decimal places?

Year 10, Chapter 1, 4 difficulty levels, Calculator allowedBuild a worksheet

Circumference

Circumference measures one complete boundary of a circle. It uses length units

  1. Identify the given length A diameter crosses the whole circle. A radius runs from the centre to the circle \(\displaystyle d=2r\)
  2. Use the boundary formula Multiply the diameter by π, or multiply the radius by two and π \(\displaystyle C=\pi d=2\pi r\)
  3. Work backwards Divide a circumference by π to find the diameter, or by two π to find the radius \(\displaystyle \begin{gathered}d=\frac{C}{\pi}\\r=\frac{C}{2\pi}\end{gathered}\)
  4. Round at the end For n complete turns without slipping, distance D is n circumferences. Use the calculator π value and round only the final distance \(\displaystyle D=nC\)

Worked example

A circle has exact circumference 15π cm. Find its radius

\[C=15\pi\text{ cm}\]
  1. \(\displaystyle 2\pi r=15\pi\)Write the circumference formula with the given exact value
  2. \(\displaystyle 2r=15\)Cancel the common non-zero factor π
  3. \(\displaystyle r=7.5\text{ cm}\)Divide by two. Check that twice this radius is the diameter 15 cm

Common mistakes

Using a diameter as a radius
✕\(\displaystyle C=2\pi(14)\)✓\(\displaystyle C=14\pi\text{ cm}\)
For diameter 14 cm, multiply by π once
Rounding each turn too early
✕\(\displaystyle D=12(2.20)=26.40\text{ m}\)✓\(\displaystyle \begin{gathered}D=12(0.7\pi)\\D\approx26.39\text{ m}\end{gathered}\)
The rounded one-turn value is not exact. Keep π until the final distance
Confusing radius with diameter
✕\(\displaystyle r=\frac{C}{\pi}\)✓\(\displaystyle r=\frac{C}{2\pi}\)
C divided by π is the diameter, twice the radius

Difficulty levels

introCalculate circumference from a given radius
standardCalculate circumference from a given diameter
challengeFind a radius from an exact circumference
examRecognise the circumference at the stated precision

Questions about this topic

Should I halve a diameter first?+

You may halve it and use 2π r, or use π d directly. Both give the same circumference

Why are the units not squared?+

Circumference is a length around the boundary, not an area inside it

Is π equal to 3.14?+

No. 3.14 is an approximation. Use the calculator π value unless a question explicitly supplies another approximation

How does a full wheel turn relate to distance?+

Without slipping, one complete turn travels one circumference. Multiply by the number of complete turns before rounding

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 1: Measurement

In this chapter

Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)