Perimeter and Circumference

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.

• Perimeter is the total distance around a shape's edge.
• It is measured in length units like cm, m and mm.
• Perimeter never uses square units.
• Area is the space inside; perimeter is the edge around it.

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.

• Perimeter of a straight-sided shape = sum of all sides.
• Rectangle: P = 2 × (length + width).
• Square: P = 4 × side; equilateral triangle: P = 3 × side.
• Find missing sides by subtracting known lengths.

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.

• The centre is the middle point of the circle.
• The radius runs from the centre to the edge.
• The diameter goes straight across through the centre.
• Diameter = 2 × radius, so d = 2r.

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.

• Pi (π) = circumference ÷ diameter, for every circle.
• Pi is about 3.14, or 3.142 for more accuracy.
• Pi's decimals never end and never repeat.
• Pi lets you find a circumference from the diameter.

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.

• Circumference C = π × d.
• With the radius, use C = 2 × π × r.
• Check first whether you have radius or diameter.
• Round the answer and give it in length units.

• Area of a circle uses the radius and π next.
• Curved compound shapes combine these ideas.
• Cylinder surface area uses the circumference.
• These skills appear in wheels, tracks and packaging.

Step 3