3.03a, 6.02d, 10.02b - Circumference and arc length

The circumference formula

A circular flower bed has radius $3\,\mathrm{m}$. Would $6\,\mathrm{m}$ of edging go all the way round? That is its diameter, a straight distance across, so we need a different length.

For every circle, the circumference divided by the diameter is the same number, pi, written $\pi$. Its value begins $3.14159\ldots$. Thus the distance round is just over three diameters, regardless of the circle's size.

[DIAGRAM: asset_name: Circumference compared with diameter; asset_slug: circle-circumference-ratio; caption: Straightening the curved boundary gives a length just over three diameters.; description: An exact circle has diameter d marked across it. A straight reference line of length pi times d is shown at the same scale beside the circle. Tick marks at one, two and three diameters show that the full circumference extends a little beyond three diameters.]

The two forms are the same rule: replacing $d$ by $2r$ gives $\pi\times2r$. For the opening bed, $C=6\pi\,\mathrm{m}$, about $18.8\,\mathrm{m}$. Six metres would cover less than a third of the boundary.

Now reverse the calculation for the same circle: its circumference is given, and the radius is unknown.

If the circumference is a decimal instead, the same division works. To find a diameter, divide by $\pi$, rather than $2\pi$.

Try the inverse calculation with a circumference that is not written using pi.

Exact answers with pi

Writing $7\pi$ keeps the exact length, while writing $21.99$ replaces it by a nearby value. Which form to use depends on the question.

Pi is irrational: its decimal expansion does not terminate or repeat. A finite decimal such as $3.14$ is an approximation to $\pi$. A calculator's pi key gives a much more accurate approximation, but an instruction to give an answer “in terms of $\pi$” asks you to keep the symbol in your final answer.

For instance, $3\pi+5\pi=8\pi$, just as three lengths plus five equal lengths make eight of them. Dividing $8\pi$ by $2$ gives $4\pi$. Pi is a number, so ordinary multiplication, addition and cancellation rules apply.

Change only the number of laps. The track stays the same, so each lap contributes the same exact length.