Perimeter is a term for the distance around a two-dimensional shape. To calculate the perimeter of any polygon (straight-sided shape) we simply add up all the lengths of the sides.
Triangles, rectangles and squares
To find the perimeter of this triangle we add up the three side lengths. Notice that all sides have been measured using the same units, so we can add them without having to first do any unit conversion.
| Perimeter | $=$= | $14+12+6$14+12+6 cm |
| $=$= | $32$32 cm |
We can save time when we carry out calculations if the shape we are looking at has side lengths that are equal.
Here is a rectangle. We know that rectangles have opposite sides of equal length.
| Perimeter | $=$= | $2\times13+2\times6$2×13+2×6 |
We have two $13$13 mm sides and two $6$6 mm sides. |
| $=$= | $26+12$26+12 mm | ||
| $=$= | $38$38 mm |
With a square, we can be even more clever. A square has $4$4 sides of the same length, so the perimeter of a square will be $4$4 times the length of one side.
| Perimeter | $=$= | $4\times7.4$4×7.4 cm |
| $=$= | $29.6$29.6 cm |
Other polygons
All perimeters can be found by adding up one side at a time 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 m |
| $=$= | $7.6$7.6 m |
We can group together sides of the same lengths to simplify calculations.
| Perimeter | $=$= | $3\times1+2\times3.6+1.4+1.5+3.45$3×1+2×3.6+1.4+1.5+3.45 m |
| $=$= | $3+7.2+2.9+3.45$3+7.2+2.9+3.45 m | |
| $=$= | $16.55$16.55 m |
To find the perimeter of a polygon, just add up all the side lengths.
Here are some shortcuts for finding the area 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
Practice questions
Question 1
Find the perimeter of the trapezium shown.
Question 2
Find the perimeter of the shape.
Question 3
A rectangle has a perimeter of $42$42 cm. If its width is $7$7 cm, what is its length?
Question 4
Find the length of wire needed to create the frame of this rectangular prism.
Units and accuracy
Perimeter is a measure of length, so we need to make sure we state the value of the perimeter together with its units. Remember that common units for lengths are 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 credit card, 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 specific unit is given in the question.
Another good practice is to perform all calculations that involve rounding right at the end (such as when $\pi$π is involved). Doing steps that involve rounding too early can result in errors that compound through the rest of the calculation, so we should save the rounding until the end.
Calculations to find measurements may involve the use of square roots (such as when using Pythagoras' theorem), the use of $\pi$π (as we have seen for arc lengths) or fractions that do not have simple decimal representations (such as $\frac{1}{7}$17).
Thus, when exact answers are required this may entail giving a response such as $\sqrt{2}$√2 m, or $3\pi$3π cm, or $\frac{5}{11}$511 km.
However, since it would not be practical to ask for $\sqrt{2}$√2 m of wood from a hardware store we often round answers for applied contexts. Giving an approximate answer by rounding to a practical number of decimal places for the given situation. For example, $\sqrt{2}$√2 m is approximately $1.41$1.41 m.