Perimeter

Perimeter is the total distance around the outside of a 2-D shape. Imagine walking once around the edge — the total distance you walk is the perimeter.

Because perimeter is a length, it uses length units: $\text{mm}$, $\text{cm}$, $\text{m}$, $\text{km}$. There are no squared or cubed units in a perimeter answer.

For straight-edged shapes the method is the same: add up all the side lengths. There are shortcuts for the regular shapes you see most often.

For a rectangle of length $l$ and width $w$: $P = 2(l + w) = 2l + 2w.$

A square is a rectangle with all four sides equal, so: $P = 4s.$

For a regular polygon with $n$ equal sides each of length $s$: $P = n \times s.$

For any straight-edged shape the method is the same: add up every side.

For a scalene triangle (no equal sides), simply add the three sides: $P = a + b + c.$

A shape's side facts can save you work. An isosceles triangle has two equal sides, so a side written once might apply twice. An equilateral triangle has all three sides equal, giving $P = 3s$.

For shapes with more sides — pentagons, hexagons and so on — write down every side length and add them in groups, or use the regular-polygon shortcut $P = n \times s$ when all sides are equal.

If the question gives you the perimeter and asks for a side length, simply work backwards. For a regular hexagon with $P = 48\text{ cm}$: $s = 48 \div 6 = 8\text{ cm}$.

A compound shape is built from two or more simple shapes. Its perimeter is the total distance around its outside, but some side lengths are not always marked.

A reliable method:

1. Find every missing side length by comparing with the marked lengths.
2. Walk around the outside of the shape, adding each side as you pass it.
3. Check: have you added the same number of lengths as the shape has sides?

For an L-shape, the two horizontal sides must add up to the same total as the longest horizontal side, and similarly for the vertical sides. So if the longest horizontal length is $12\text{ cm}$ and the marked top is $8\text{ cm}$, the missing horizontal step is $12 - 8 = 4\text{ cm}$.

A neat shortcut for rectilinear L-shapes (all corners are $90\degree$): the total horizontal travel equals twice the longest horizontal side; the total vertical travel equals twice the longest vertical side. So the perimeter is $2 \times \text{longest horizontal} + 2 \times \text{longest vertical}$.

The perimeter of a circle has its own name: the circumference.

Every circle has the same ratio of circumference to diameter — that ratio is the famous number pi, written $\pi$. Its decimal value starts $3.14159\ldots$, and most calculators have a $\pi$ button.

So for a circle of diameter $d$ (or radius $r$, where $d = 2r$): $C = \pi d \quad \text{or equivalently} \quad C = 2\pi r.$

A circle of diameter $14\text{ cm}$ has circumference $C = \pi \times 14 \approx 44.0\text{ cm}$ (to 3 significant figures).

Some questions want an exact answer with $\pi$ left in. Don't multiply it out — write $14\pi\text{ cm}$. Read the question to see whether a decimal or an answer in terms of $\pi$ is expected.

Working backwards is just as useful. If a circle has $C = 50\text{ cm}$, then $d = 50 \div \pi \approx 15.9\text{ cm}$ and $r = d \div 2 \approx 7.96\text{ cm}$.

An arc is part of the edge of a circle. The piece of pie shape it bounds (with two radii) is a sector, and the angle at the centre of the sector is the sector angle, often called $\theta$.

The full circle has a centre angle of $360\degree$, so an arc with centre angle $\theta$ is the fraction $\dfrac{\theta}{360}$ of the whole circumference: $\text{arc length} = \frac{\theta}{360} \times \pi d.$

A sector with centre angle $90\degree$ in a circle of radius $10\text{ cm}$ has: $\text{arc} = \frac{90}{360} \times \pi \times 20 = \frac{1}{4} \times 20\pi = 5\pi \approx 15.7\text{ cm}.$

A semicircle is a special case: $\theta = 180\degree$, so the arc is exactly half the circumference. The full perimeter of a semicircle is half-circumference plus diameter (the straight edge across the bottom).

Real-life shapes — a running track, a stadium plan, an athletics field — often mix straight edges and curved edges. The perimeter is still the total distance around the outside; you just calculate each piece separately and add them together.

For the track shape: two straight sides of $100\text{ m}$ each give $200\text{ m}$, and two semicircles of radius $30\text{ m}$ make exactly one full circle with $C = 2\pi \times 30 = 60\pi \approx 188.5\text{ m}$. The total perimeter is $200 + 60\pi \approx 388.5\text{ m}$.

The plan stays the same whatever the shape: list the parts, calculate each piece, then add them up.

Perimeter is the first step on a long road through geometry:

• Area and surface area use the same shapes and the same care with units.
• Volume and cross-sections of 3-D shapes use perimeters of 2-D cross-sections.
• Pythagoras' theorem and trigonometry rely on knowing the sides of right-angled triangles — same care, more powerful tools.
• Real-life planning — fencing a garden, framing a picture, edging a flower bed, designing a running track — all uses perimeter directly.

If a perimeter question feels tricky, the fix is almost always to slow down, list every side, find any missing ones, and only then add. Keep this revision close — you'll come back to these methods again and again.