# 4.02 Calculating perimeter

Lesson

The word 'perimeter' comes from the Greek word perimetron, where peri means 'around' and metron means 'measure'.

Therefore perimeter is defined as follows:

Perimeter

Perimeter is the distance around a two-dimensional shape.

It is equal to the sum of the lengths of all sides of the shape.

## Calculating perimeter

To find the perimeter of any polygon (straight sided shape) we add together the lengths of all the sides that form the boundary of the shape.

### Triangle

Here is a scalene triangle. To find its perimeter we add together the three side lengths. Notice that all sides are measured using the same units.

 Perimeter $=$= $14+12+6$14+12+6 $=$= $32$32 cm

Of course, we can get clever with the way we carry out calculations, if the shape has side lengths that are equal. We mark sides that are equal with matching dashes.

Here is a rectangle. We know that rectangles have opposite sides of equal length, so we can double the side lengths in our calculations.

 Perimeter $=$= $2\times13+2\times6$2×13+2×6 $=$= $26+12$26+12 $=$= $38$38 mm

With a square, we can be even more clever. A square has $4$4 sides of equal length, so the perimeter of a square will be $4$4$\times$× the side length.

 Perimeter $=$= $4\times7.4$4×7.4 $=$= $29.6$29.6 cm

Remember!

To find the perimeter of a shape, we add together all the side lengths. Here are some speedy shortcuts for finding the perimeter of a rectangle and a square.

 Perimeter of a rectangle $=$= $2\times\left(\text{length }+\text{width }\right)$2×(length +width ) Perimeter of a square $=$= $4\times\text{side length }$4×side length

### Other shapes

All perimeters can be found by adding together each side length as we travel around the shape.

Here is a trapezium. To find its perimeter we add up the side lengths.

 Perimeter $=$= $1.2+1.3+2.4+2.7$1.2+1.3+2.4+2.7 $=$= $7.6$7.6 m

We can often use simple rules to add together sides of the same length, or to determine the lengths of sides that don't have measurements. For example,

 Perimeter $=$= $3\times1+2\times3.6+1.4+1.5+3.45$3×1+2×3.6+1.4+1.5+3.45 $=$= $3+7.2+6.35$3+7.2+6.35 $=$= $16.55$16.55 m

### Units

Because perimeter is a measure of length, we need to make sure we use the units we are given. Remember from our lesson on units of length, that common units include millimetres (mm), centimetres (cm), metres (m) and kilometres (km).

All of these could be used to measure the perimeter of different sized objects. Millimetres could be used to measure the perimeter of a sim card, centimetres to measure the perimeter of a wallet, metres to measure the perimeter of a room and kilometres to measure the perimeter of a town.

It is good mathematical practice to use the word 'units', if no particular unit is given.

#### Worked example

Find the perimeter of the figure below.

Notice that we can't immediately add all of the sides together because we don't know the length of side $AD$AD, which is marked as $x$x cm.

(a) First, find the value of $x$x.

Think: Divide the figure into a rectangle and a right-angled triangle by constructing a line from vertex $D$D to meet side $AB$AB at a right angle. This line will be $12$12 cm in length, the same as side $BC$BC. It creates a right-angled triangle with $x$x as the hypotenuse, and shorter sides of length $5$5 cm and $12$12 cm. We can now use Pythagoras' theorem to find the value of $x$x .

Do:

 $c^2$c2 $=$= $a^2+b^2$a2+b2 $x^2$x2 $=$= $5^2+12^2$52+122 (Substitute $c=x$c=x, $a=5$a=5 and $b=12$b=12) $x^2$x2 $=$= $25+144$25+144 $x^2$x2 $=$= $169$169 $x$x $=$= $\sqrt{169}$√169 (Take the square root of both sides) $x$x $=$= $13$13 cm

(b) Find the perimeter.

Think: We can now add the length of the sides together to find the perimeter.

Do:

 Perimeter $=$= $13+10+12+15$13+10+12+15 $=$= $50$50 cm

#### Practice questions

##### Question 1

Find the perimeter of the rectangle shown.

##### Question 2

Find the side length $d$d indicated on the diagram. The perimeter of the shape is $35$35 cm.

##### Question 3

Find the perimeter of the shape.

##### Question 4

Find the perimeter of the kite shown in millimetres.

### Outcomes

#### MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures