Year 9 Maths Measurement

Length and Perimeter

Q: What is the formula for the perimeter of a rectangle?

P = 2l + 2w, or P = 2(l + w): twice the length plus twice the width.

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Length and Perimeter - Video - An Introduction to Perimeter Watch

Theory

The perimeter is the distance all the way round a shape. This Year 9 maths page revises the Year 8 work: adding sides in the same unit, using marks for equal sides, finding missing sides, writing perimeters with pronumerals, and working backwards in fencing problems. Curriculum: VC2M8M01 / AC9M8M01 (Year 8, revised).

The perimeter of a shape is the total length of its outside edge. Start at a corner, walk round once and add every side. Perimeter is a length, so it is measured in mm, cm, m or km.

Sides with the same tick marks are equal. A regular polygon has all its sides equal, so its perimeter is the number of sides times one side.

All the sides must be in the same unit before you add them. $1$ cm $= 10$ mm, $1$ m $= 100$ cm and $1$ km $= 1000$ m.

In a shape whose corners are all right angles, a missing side comes from the sides opposite it: the pieces going one way add up to the full length on the other side.

A perimeter can be written as an expression when a side is a pronumeral. If the perimeter is known, the expression makes an equation to solve.

With only right angles, the sides going one way balance: the bottom is $a + c$ and the left side is $d = e + f$.

To a smaller unit, multiply (there are more of them). To a bigger unit, divide. Change every side to one unit before adding.

\[P_{\text{rectangle}} = 2l + 2w = 2(l + w) \qquad P_{\text{square}} = 4s\]

P = 2 (l + w)

A regular polygon with $n$ sides, each of length $s$:

\[P = n \times s \qquad s = \dfrac{P}{n}\]

P = n \times s

Change	Rule	Example
cm to mm	$\times 10$	$4.5 \text{ cm} = 45 \text{ mm}$
m to cm	$\times 100$	$0.84 \text{ m} = 84 \text{ cm}$
km to m	$\times 1000$	$2.3 \text{ km} = 2300 \text{ m}$
mm to m	$\div 1000$	$650 \text{ mm} = 0.65 \text{ m}$

A slanted side. If a side is slanted and not given, draw a right-angled triangle under it and use Pythagoras. A right trapezium with parallel sides \(10\) and \(16\) and height \(8\) has a slanted side of \(\sqrt{6^2 + 8^2} = 10\), so its perimeter is \(10 + 16 + 8 + 10 = 44\).

Steps and staircases. If every corner is a right angle and the shape only steps inwards at one corner, the small steps add up to the full width and the full height. Its perimeter is the same as the rectangle around it.

Fences and wire. A side along a wall is not fenced. A gate is left out of the fence length. Several strands of wire multiply the length. Fencing is bought in whole panels or rolls, so always round up.

How to find a perimeter

Change every length to the same unit.
Fill in unmarked sides: equal tick marks, sides opposite in a right-angled shape, or Pythagoras for a slanted side.
Walk round the outside once from a corner, adding each side. Lines inside the shape do not count.
Write the unit: mm, cm, m or km.

Working backwards

Write the perimeter as an expression, using a pronumeral for the unknown side.
Set it equal to the perimeter you are given and solve the equation.
Check by putting your answer back in and adding the sides.

Example 1 — Equal sides and mixed units

(i) An isosceles triangle has a base of $4.5$ cm and two equal sides of $62$ mm. Find its perimeter in millimetres.
(ii) A regular hexagon has a perimeter of $0.84$ m. Find the length of one side in centimetres.

Solution

(i) Change the base to millimetres first: $4.5 \text{ cm} = 45 \text{ mm}$. The two equal sides are both $62$ mm.

$P$	$=$	$45 + 62 + 62$
	$=$	$\mathbf{169} \text{ mm}$

(ii) Change to centimetres: $0.84 \text{ m} = 84 \text{ cm}$. A hexagon has $6$ equal sides, so share the perimeter between them.

$s$	$=$	$84 \div 6$
	$=$	$\mathbf{14} \text{ cm}$

s = 84 \div 6 = 14

Adding $4.5 + 62 + 62$ without changing units would give a meaningless answer.

Example 2 — Missing sides, then the perimeter

An L-shape has right-angled corners. Its bottom is $14$ m and its left side is $9$ m. A step $6$ m wide and $4$ m deep is missing from the top right. Find the perimeter.

Solution

Find the two missing sides first. The two top pieces must add to the $14$ m bottom. The two right pieces must add to the $9$ m left side.

$\text{top}$	$=$	$14 - 6 = 8 \text{ m}$
$\text{right}$	$=$	$9 - 4 = 5 \text{ m}$

Now walk round the outside once.

$P$	$=$	$14 + 5 + 6 + 4 + 8 + 9$
	$=$	$\mathbf{46} \text{ m}$

P = 14 + 5 + 6 + 4 + 8 + 9 = 46

Example 3 — An expression, then an equation

A rectangle is three times as long as it is wide. Let its width be $w$ cm.
(i) Write an expression for its perimeter.
(ii) The perimeter is $96$ cm. Find the length and the width.

Solution

(i) The length is $3w$. Add all four sides and collect like terms.

$P$	$=$	$w + 3w + w + 3w$
	$=$	$\mathbf{8w} \text{ cm}$

(ii) Make the expression equal to $96$ and solve.

$8w$	$=$	$96$
$w$	$=$	$12$

8 w = 96

The width is $12$ cm and the length is $3 \times 12 =$ $36$ cm. Check: $2 \times (36 + 12) = 96$ $\checkmark$

Example 4 — Fencing with a gate

A rectangular vegetable garden is $14.5$ m long and $6$ m wide. It is fenced on all four sides, except for a $0.9$ m gate. Fencing is sold in panels $2.4$ m long at $45 each. How many panels are needed, and what do they cost?

Solution

Find the perimeter, then take away the gate.

$P$	$=$	$2 \times (14.5 + 6) = 41 \text{ m}$
$\text{fence}$	$=$	$41 - 0.9 = 40.1 \text{ m}$

Divide by the length of one panel. A part panel still has to be bought, so round up.

$40.1 \div 2.4$	$=$	$16.708\ldots$
$\text{panels}$	$=$	$\mathbf{17}$
$\text{cost}$	$=$	$17 \times \$45 = \mathbf{\$765}$

40.1 \div 2.4 \approx 16.7

Rounding down to $16$ panels would leave a gap of almost $2$ m.

Common pitfalls

Change to one unit first. Sides of $2.5$ m and $80$ cm add to $250 + 80 = 330$ cm, not $82.5$.

Only the outside edge counts. A line drawn inside a shape to split it is not part of the perimeter.

A cut-out can make the perimeter bigger. Cutting a notch out of an edge removes one short length but adds the sides of the notch.

Find every missing side. A shape with right-angled corners can have two or more unmarked sides. Count the sides as you walk round to make sure none is missed.

Round up for things you buy. If $16.7$ rolls or panels are needed, buy $17$.

Frequently asked questions

What is perimeter?

Perimeter is the total distance around the outside edge of a shape. Add the lengths of all its sides, in the same unit.

What is the formula for the perimeter of a rectangle?

$P = 2l + 2w$, or $P = 2(l + w)$: twice the length plus twice the width.

How do you find the side of a regular polygon from its perimeter?

Divide the perimeter by the number of sides. A regular hexagon with a perimeter of $84$ cm has sides of $14$ cm.

How do you find a missing side of a shape with right angles?

Use the sides opposite it. The lengths going one way add up to the full length on the opposite side. If the bottom is $14$ m and one top piece is $6$ m, the other top piece is $8$ m.

How do you find a side when you know the perimeter?

Write the perimeter as an expression with a pronumeral for the unknown side, make it equal to the perimeter and solve the equation.

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Circumference and Perimeter of a Sector

Change	Rule	Example
cm to mm	\(\times 10\)	\(4.5 \text{ cm} = 45 \text{ mm}\)
m to cm	\(\times 100\)	\(0.84 \text{ m} = 84 \text{ cm}\)
km to m	\(\times 1000\)	\(2.3 \text{ km} = 2300 \text{ m}\)
mm to m	\(\div 1000\)	\(650 \text{ mm} = 0.65 \text{ m}\)

\(P\)	\(=\)	\(45 + 62 + 62\)
\(\)	\(=\)	\(\mathbf{169} \text{ mm}\)

\(s\)	\(=\)	\(84 \div 6\)
\(\)	\(=\)	\(\mathbf{14} \text{ cm}\)

\(\text{top}\)	\(=\)	\(14 - 6 = 8 \text{ m}\)
\(\text{right}\)	\(=\)	\(9 - 4 = 5 \text{ m}\)

\(P\)	\(=\)	\(14 + 5 + 6 + 4 + 8 + 9\)
\(\)	\(=\)	\(\mathbf{46} \text{ m}\)

\(P\)	\(=\)	\(w + 3w + w + 3w\)
\(\)	\(=\)	\(\mathbf{8w} \text{ cm}\)

\(P\)	\(=\)	\(2 \times (14.5 + 6) = 41 \text{ m}\)
\(\text{fence}\)	\(=\)	\(41 - 0.9 = 40.1 \text{ m}\)

\(40.1 \div 2.4\)	\(=\)	\(16.708\ldots\)
\(\text{panels}\)	\(=\)	\(\mathbf{17}\)
\(\text{cost}\)	\(=\)	\(17 \times \$45 = \mathbf{\$765}\)

\(8w\)	\(=\)	\(96\)
\(w\)	\(=\)	\(12\)

Length and Perimeter

📖 Theory

How to find a perimeter

Working backwards

Common pitfalls

Frequently asked questions

Theory