Perimeter and Circumference

Perimeter is the total distance around the outside edge of a flat shape. Picture an ant walking all the way around the border and back to where it started — the distance it travels is the perimeter.

Because perimeter is a distance, it is measured in ordinary length units — centimetres ($\text{cm}$), metres ($\text{m}$) or millimetres ($\text{mm}$). It is never in square units; those belong to area.

You meet perimeter all the time:

• The length of fencing needed to go around a garden.
• The amount of ribbon to wrap once around a present.
• The distance you run around the edge of a sports field.
• The frame length needed for a picture.

Keep the difference clear in your head: perimeter is the journey around the edge, while area is the space covered inside.

For any shape with straight sides, the perimeter is found by adding up every side.

For the pentagon above: $P = 6 + 5 + 7 + 4 + 8 = 30\text{ cm}.$

Some shapes give you helpful shortcuts:

• A rectangle has two lengths and two widths, so $P = 2 \times (\text{length} + \text{width})$. A $9\text{ cm}$ by $4\text{ cm}$ rectangle has perimeter $2 \times (9 + 4) = 26\text{ cm}$.
• A square has four equal sides, so $P = 4 \times \text{side}$.
• An equilateral triangle has three equal sides, so $P = 3 \times \text{side}$.

Two things to watch:

• Make sure every length is in the same unit before you add.
• For an L-shape or other compound shape, some side lengths may be missing — work them out by subtracting known lengths first.

A circle is a perfectly round shape where every point on the edge is the same distance from the middle. To work with circles you need the right words.

• The centre is the point in the exact middle.
• The radius is the distance from the centre to the edge.
• The diameter is the distance straight across, passing through the centre.
• The circumference is the distance all the way around the edge — it is the circle's perimeter.

The radius and diameter are linked by a simple rule: $\text{diameter} = 2 \times \text{radius}, \qquad d = 2r.$

So if a circle has radius $5\text{ cm}$, its diameter is $10\text{ cm}$. If you are told the diameter is $18\text{ cm}$, the radius is $9\text{ cm}$ — just halve it.

Here is something remarkable. Take any circle, big or small, and divide its circumference by its diameter. You always get the same number — about $3.14$. This number is called pi, written with the Greek letter $\pi$.

So $\pi$ tells you how many diameters fit around the circumference — a little more than three of them.

A few facts about $\pi$:

• It is roughly $3.14$, or $3.142$ for more accuracy.
• Its decimals go on forever without repeating, so we always round it.
• Many calculators have a $\pi$ button that uses lots of decimal places.

Because $\pi$ never changes, it gives us a powerful shortcut: if you know a circle's diameter, you can work out its circumference without measuring around the edge at all.

Because the circumference divided by the diameter is always $\pi$, you can rearrange that into a formula for the circumference: $C = \pi \times d.$

So for a circle with diameter $8\text{ cm}$, using $\pi \approx 3.14$: $C = 3.14 \times 8 = 25.12\text{ cm}.$

Often you are given the radius instead. Since the diameter is twice the radius, the formula becomes: $C = 2 \times \pi \times r.$

A circle with radius $5\text{ cm}$ has circumference $2 \times 3.14 \times 5 = 31.4\text{ cm}$.

A reliable method:

1. Check whether you have the radius or the diameter.
2. If you have the radius, double it to get the diameter (or use $C = 2\pi r$).
3. Multiply by $\pi$.
4. Round your answer sensibly and add the length units.

Getting confident with perimeter and circumference opens up plenty ahead:

• Area of a circle — next you will meet $A = \pi r^2$, which reuses the radius and $\pi$.
• Compound and circular shapes — semicircles and shapes with curved edges combine perimeter and circumference ideas.
• 3D solids — the circumference of a circle helps find the surface area of cylinders.
• In everyday life, you will use these skills for fencing, wheels, tracks, pipes and packaging.

If a measurement topic later feels hard, it is often a perimeter or circle skill underneath that needs a quick refresh. Keep this guide nearby — circles get friendlier with every question.