Perimeter and Circumference

Perimeter is the total distance around the outside edge of a flat shape. Picture walking all the way around the border and back to your start — that distance is the perimeter.

For a shape with straight sides, the perimeter is simply the sum of all the sides. Some shapes give shortcuts:

• A rectangle: $P = 2 \times (\text{length} + \text{width})$.
• A square: $P = 4 \times \text{side}$.

Because perimeter is a distance, it uses ordinary length units — $\text{cm}$, $\text{m}$, $\text{mm}$ — and never square units; those belong to area.

This year you push further: compound shapes with hidden sides, and shapes with curved edges where you bring in the circumference. The same idea always holds — perimeter is the whole journey around the boundary.

A compound shape is made by joining simple shapes together — an L-shape, a T-shape, or a rectangle with a piece cut out. The challenge is that some side lengths are not labelled: you have to work them out.

Here is a reliable method:

1. Find the missing sides. Opposite parts of a compound shape line up, so a missing length equals one labelled length minus another. In the diagram, the short top step is $18 - 12 = 6$, and the inner vertical side is $13 - 6 = 7$.
2. Trace the whole boundary with your finger, listing every side as you go.
3. Add them all up — and keep every length in the same unit.

For this L-shape: $P = 12 + 7 + 6 + 6 + 18 + 13 = 62$.

The most common slip is missing a side completely, so always trace right around and count every edge.

The circumference is the distance around a circle — its perimeter. It is linked 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 diameter $8\text{ cm}$ has circumference $\pi \times 8 \approx 25.1\text{ cm}$.

This year you also use the formula in reverse. If you know the circumference and want the diameter, rearrange it: $d = \frac{C}{\pi}.$

So a circle with circumference $31.4\text{ cm}$ has diameter $31.4 \div \pi \approx 10\text{ cm}$, which makes the radius $5\text{ cm}$.

The key habit: decide first what you have (circumference, radius or diameter) and what you want, then pick the matching version of the formula.

A semicircle is half a circle. Its perimeter is not just half the circumference — you must also include the straight edge, the diameter.

To find the perimeter of a semicircle:

1. Work out the whole circumference: $C = \pi \times d$.
2. Halve it for the curved part.
3. Add the diameter for the straight edge.

For the semicircle above, with diameter $12\text{ cm}$: $\text{curved part} = \frac{\pi \times 12}{2} \approx 18.8\text{ cm}, \qquad P \approx 18.8 + 12 = 30.8\text{ cm}.$

The same idea works for any shape with curved edges — a running track, a window with a rounded top, a pond. Split the boundary into straight parts and curved parts, work each out, then add them together.

An arc is part of the circumference of a circle — like a slice of crust around a sector. A sector is the pizza-slice shape between two radii.

The clever idea: an arc is the same fraction of the whole circumference as its angle is of the full $360\degree$.

So the arc length formula is: $\text{arc length} = \frac{\text{angle}}{360\degree} \times C = \frac{\text{angle}}{360\degree} \times \pi d.$

For a circle of diameter $20\text{ cm}$ with a $90\degree$ sector:

1. Full circumference: $C = \pi \times 20 \approx 62.8\text{ cm}$.
2. The fraction is $\frac{90}{360} = \frac{1}{4}$.
3. Arc length $= \frac{1}{4} \times 62.8 \approx 15.7\text{ cm}$.

Quick check: a half circle uses angle $180\degree$, giving the fraction $\frac{1}{2}$ — exactly the curved part of a semicircle you met earlier.

Getting confident with perimeter and circumference opens up plenty ahead:

• Area of circles and sectors — next you will meet $A = \pi r^2$, and the area of a sector uses the same fraction idea as arc length.
• Surface area of solids — the circumference helps find the surface area of cylinders.
• Compound and curved shapes — combining straight and curved parts is a skill you will use often.
• In everyday life, these skills appear in running tracks, wheels, pipes, fencing and packaging design.

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 and compound shapes get friendlier with every question.