Perimeter
Every question with a fully worked solution.
- Perimeter - Video - Perimeter Watch
Theory
The perimeter of a shape is the distance all the way around its outside. To find it, add the lengths of all the sides. This Year 7 maths page shows the shortcuts for rectangles, squares and other four-sided shapes, how to find missing sides on composite shapes, and how to work backwards from a perimeter.
The perimeter is the distance around a shape: add up all the side lengths. It is a length, so its units are mm, cm, m or km.
Tick marks: sides with the same marks are the same length. Opposite sides of a rectangle and a parallelogram are equal. All four sides of a square or a rhombus are equal. A kite has two pairs of equal sides next to each other.
Rectangle: \(P = l + w + l + w = 2l + 2w = 2(l + w)\). Square or rhombus: \(P = 4s\). Parallelogram or kite: \(P = 2a + 2b\). Trapezium: add all four sides.
Composite shape. Find each missing side first, using the sides opposite it. Then start at a corner and go round once, adding every side.
Push the sides out. When every corner is a right angle, a shape with a corner cut out has the same perimeter as the rectangle around it.
Working backwards. From the perimeter, take away the sides you know, then share what is left between the equal sides.
Perimeter formulas:
| Shape | Perimeter |
|---|---|
| rectangle | \(P = 2l + 2w\) |
| square, rhombus | \(P = 4s\) |
| parallelogram, kite | \(P = 2a + 2b\) |
| any other shape | add every side |
How to find the perimeter of a composite shape
- Find each missing side from the sides opposite it.
- Start at a corner and go round once, writing every side.
- Add them, and check by pushing the sides out if all corners are right angles.
- Write the units (one unit for every side).
| \(DE\) | \(=\) | \(8 - 3 = 5\) |
| \(DC\) | \(=\) | \(12 - 3 = 9\) |
| \(P\) | \(=\) | \(12 + 3 + 9 + 5 + 3 + 8\) |
| \(\) | \(=\) | \(40\) |
The perimeter is \(40\) units.
| \(AB\) | \(=\) | \(3 + 4 + 3 = 10\) |
| \(P\) | \(=\) | \(10 + 3 + 3 + 4\) |
| \(\) | \(\) | \(+\, 4 + 4 + 3 + 3\) |
| \(\) | \(=\) | \(34\) |
\(AH = BC = 3\) and \(GF = DE = 4\). The perimeter is \(34\) units.
| \(a\) | \(=\) | \(10 - 4 = 6\) |
| \(b\) | \(=\) | \(8 - 3 = 5\) |
| \(P\) | \(=\) | \(10 + 3 + 6 + 5 + 4 + 8\) |
| \(\) | \(=\) | \(36\) |
\(36\) cm. Check by pushing the sides out: \(2 \times 10 + 2 \times 8 = 36\).
| \(0.8 \times 100\) | \(=\) | \(80\) |
| \(45 + 45 + 80 + 80\) | \(=\) | \(250\) |
\(250\) cm of tape
(i) Find the perimeter.
(ii) How much fencing is needed?
(iii) What does the fencing cost?
| \(2 \times 60 + 2 \times 35\) | \(=\) | \(190\) |
| \(190 - 4\) | \(=\) | \(186\) |
| \(186 \times 12\) | \(=\) | \(2232\) |
(i) \(190\) m
(ii) \(186\) m
(iii) \(\$2232\)
(ii) A right trapezium has parallel sides \(9\) m and \(6\) m, height \(4\) m and slanted side \(5\) m. Find its perimeter.
| \(6 + 3 + 3 + 4 + 4\) | \(=\) | \(20\) |
| \(9 + 6 + 4 + 5\) | \(=\) | \(24\) |
(i) \(20\) m (the top of the rectangle is inside, so it is not counted)
(ii) \(24\) m
(ii) A rhombus has a perimeter of \(26\) cm. How long is each side?
| \(27 \div 6\) | \(=\) | \(4.5\) |
| \(2 \times 4.5\) | \(=\) | \(9\) |
| \(26 \div 4\) | \(=\) | \(6.5\) |
(i) A length is \(2\) widths, so the perimeter is \(6\) widths: width \(4.5\) m, length \(9\) m. Check: \(9 + 4.5 + 9 + 4.5 = 27\).
(ii) \(6.5\) cm
Common pitfalls
Frequently asked questions
What is perimeter?
The distance all the way around the outside of a shape.
What is the perimeter formula for a rectangle?
\(P = 2l + 2w\), or \(2(l + w)\).
How do you find the perimeter of a square?
\(P = 4s\): four times one side.
How do you find a missing side on a composite shape?
Use the sides opposite it: the total across one way equals the total across the other way.
What units does perimeter have?
Units of length, such as cm or m.