Perimeter and Circumference

The circumference is the perimeter of a circle — the distance once around its edge. It is tied to the diameter by the special number pi, $\pi \approx 3.14$: $C = \pi \times d, \qquad C = 2 \times \pi \times r.$

So a circle with radius $7\text{ cm}$ has diameter $14\text{ cm}$ and circumference $\pi \times 14 \approx 44.0\text{ cm}$.

You can also use the formula in reverse. If you know the circumference, the diameter is $d = \dfrac{C}{\pi}$, and halving that gives the radius.

This year you push further: compound shapes that mix straight and curved edges, the perimeter of a sector, and multi-step problems where arc length is one part of a longer calculation. The first habit, every time, is to decide whether you have been given the radius or the diameter.

An arc is a part of the circumference of a circle. A sector is the pizza-slice shape between two radii, and the arc is its curved edge.

The clever idea: an arc is the same fraction of the whole circumference as its angle is of the full $360\degree$. $\text{arc length} = \frac{\text{angle}}{360\degree} \times C = \frac{\text{angle}}{360\degree} \times \pi d.$

For a circle of radius $9\text{ cm}$ with a $120\degree$ sector:

1. Full circumference: $C = 2 \times \pi \times 9 \approx 56.5\text{ cm}$.
2. The fraction is $\dfrac{120}{360} = \dfrac{1}{3}$.
3. Arc length $= \dfrac{1}{3} \times 56.5 \approx 18.8\text{ cm}$.

A quick check: an angle of $180\degree$ gives the fraction $\tfrac{1}{2}$ — the curved part of a semicircle. An angle of $90\degree$ gives a quarter circle. The fraction always matches the slice you can picture.

The perimeter of a sector is the distance all the way around its edge — and a sector has three edges: the curved arc and two straight radii.

A common slip is to give only the arc length. But trace the boundary with your finger: you walk along one radius, around the arc, and back along the other radius. So: $\text{sector perimeter} = \text{arc length} + 2 \times \text{radius}.$

Example: a sector has radius $10\text{ cm}$ and angle $90\degree$.

1. Full circumference: $C = 2 \times \pi \times 10 \approx 62.8\text{ cm}$.
2. Arc length: $\dfrac{90}{360} \times 62.8 \approx 15.7\text{ cm}$.
3. Add the two radii: $15.7 + 10 + 10 = 35.7\text{ cm}$.

The same care applies to a semicircle, which is just a $180\degree$ sector: its perimeter is the curved half plus the straight diameter.

By Stage 9 the shapes get richer: a rectangle with a semicircular end, a running track, a window with a rounded top. These compound shapes mix straight and curved edges, but one method handles them all.

The reliable approach:

1. Trace the whole boundary and split it into straight parts and curved parts.
2. Work out each straight part as an ordinary length.
3. Work out each curved part using circumference or arc length.
4. Add everything together — only the boundary, never an inside edge.

For the shape above — a $20\text{ cm}$ by $12\text{ cm}$ rectangle with a semicircle on the $12\text{ cm}$ end:

• Three straight sides: $20 + 20 + 12 = 52\text{ cm}$.
• The curved end is a semicircle of diameter $12\text{ cm}$: $\dfrac{\pi \times 12}{2} \approx 18.8\text{ cm}$.
• Total perimeter $\approx 52 + 18.8 = 70.8\text{ cm}$.

The straight side that the semicircle replaces is not part of the boundary, so it is not counted.

Stage 9 expects you to use the circle formulas in reverse — to start from a perimeter and find a missing measurement.

From a circumference to a radius. Rearrange $C = \pi d$ to $d = \dfrac{C}{\pi}$, then halve for the radius. A circle with circumference $50\text{ cm}$ has diameter $50 \div \pi \approx 15.9\text{ cm}$, so radius $\approx 8.0\text{ cm}$.

From an arc length to an angle. Rearrange the arc formula. Since $\text{arc} = \dfrac{\text{angle}}{360\degree} \times C$: $\text{angle} = \frac{\text{arc length}}{C} \times 360\degree.$

Example: a sector has radius $6\text{ cm}$ and an arc of $9.42\text{ cm}$.

1. Full circumference: $C = 2 \times \pi \times 6 \approx 37.7\text{ cm}$.
2. Angle $= \dfrac{9.42}{37.7} \times 360\degree \approx 90\degree$.

So the sector is a quarter circle. Working backwards is just careful rearranging — decide what you know, what you want, and which formula links them.

Getting confident with perimeter, circumference and arc length opens up plenty ahead:

• Area of circles and sectors — the area of a sector uses the very same fraction idea as arc length.
• Surface area and volume of solids — cylinders and cones rely on the circumference of a circle.
• Compound and curved shapes — splitting a boundary into straight and curved parts is a skill you will use again and again.
• In everyday life, these calculations appear in athletics tracks, wheels, pipes, gears and packaging design.

If a later measurement topic feels hard, it is often a circle skill underneath that needs a quick refresh. Keep this guide nearby — compound shapes and sectors get friendlier with every question you try.