Perimeter
Theory
Perimeter is the total distance around the outside of a shape. Change every side to the same unit, then add each side once. A rhombus with sides of \(2.5\) cm has a perimeter of \(4 \times 2.5 = 10\) cm. This Year 8 maths page shows how.
Perimeter is a length, so its units are mm, cm, m or km. Change every side to the same unit before you add.
Equal sides save work. Matching marks mean equal sides. A regular polygon has \(n\) equal sides, so \(P = n \times s\). Rectangle: \(P = 2l + 2w\). Square or rhombus: \(P = 4s\). Kite or parallelogram: \(P = 2a + 2b\). A trapezium has no shortcut: add all four sides.
Composite shapes. When every corner is a right angle, the pieces along one side add up to the side opposite, so find the missing sides first. Then go round once and add.
Push the steps out. A staircase of right-angled steps has the same perimeter as the rectangle around it.
A notch makes the perimeter longer. The piece of edge removed is replaced by the new cut edges.
Pronumerals. Add the sides and collect like terms: \(x + x + 3x = 5x\). Given the perimeter, solve for \(x\). Working backwards: take the known sides off the perimeter, then share what is left between the equal sides.
Then the question goes on. A wall or a gate is not fenced; cost \(=\) length \(\times\) price per metre; lengths sold in a fixed size are rounded up.
Perimeter formulas:
| Shape | Perimeter |
|---|---|
| regular polygon with \(n\) sides of \(s\) | \(P = ns\) |
| rectangle | \(P = 2l + 2w\) |
| square, rhombus | \(P = 4s\) |
| kite, parallelogram | \(P = 2a + 2b\) |
| any other shape | add every side once |
How to find a perimeter
- Change every side to the same unit.
- Find any missing sides (opposite pieces add up).
- Go round once and add every side, then answer the question asked.
(ii) A parallelogram has a perimeter of \(24\) cm, and one side is twice the length of the other. Find the shorter side.
(ii) Let the shorter side be \(a\) cm, so the longer side is \(2a\) cm. Opposite sides are equal.
| \(P\) | \(=\) | \(4 \times 2.5 = 10\) |
| \(a + a + 2a + 2a\) | \(=\) | \(24\) |
| \(6a\) | \(=\) | \(24\) |
| \(a\) | \(=\) | \(4\) |
(i) \(10\) cm
(ii) The shorter side is \(4\) cm (and the longer side \(8\) cm).
| \(320\text{ mm}\) | \(=\) | \(32\text{ cm}\) |
| \(P\) | \(=\) | \(2 \times 32 + 2 \times 55\) |
| \(\) | \(=\) | \(64 + 110 = 174\) |
\(174\) cm
The two shoulders are equal and together are the top minus the stem: \(a = (12 - 4) \div 2 = 4\). The right end equals the left end, \(2\) m.
| \(P\) | \(=\) | \(12 + 2 + 4 + 3\) |
| \(\) | \(\) | \(+ 4 + 3 + 4 + 2\) |
| \(\) | \(=\) | \(34\) |
\(34\) m. Check by pushing out to a \(12\) m by \(5\) m rectangle: \(2 \times 12 + 2 \times 5 = 34\).
(i) Write a simplified expression for the perimeter.
(ii) Find the width and the length.
Let the width be \(w\) m, so the length is \((2w + 5)\) m.
| \(P\) | \(=\) | \(2w + 2(2w + 5)\) |
| \(\) | \(=\) | \(6w + 10\) |
| \(6w + 10\) | \(=\) | \(52\) |
| \(w\) | \(=\) | \(7\) |
(i) \((6w + 10)\) m
(ii) Width \(7\) m, length \(2 \times 7 + 5 = 19\) m. Check: \(14 + 38 = 52\).
(i) Find the perimeter of the outside of the frame.
(ii) How many lengths of timber are needed?
The border adds \(5\) cm at each end of both sides: \(40 + 10 = 50\) and \(30 + 10 = 40\).
| \(P\) | \(=\) | \(2 \times 50 + 2 \times 40 = 180\) |
| \(180 \div 50\) | \(=\) | \(3.6\) |
(i) \(180\) cm
(ii) Three lengths give only \(150\) cm, so round up: \(4\) lengths.
Common pitfalls
Frequently asked questions
What is perimeter?
The total distance around the outside of a shape.
What is the perimeter of a rectangle?
\(P = 2l + 2w\): twice the length plus twice the width.
How do I find a missing side in a composite shape?
When all corners are right angles, the pieces along one side add up to the side opposite, so subtract.
What units is perimeter measured in?
Units of length, such as mm, cm, m or km.
How do I find a side from the perimeter?
Take the known sides off the perimeter, then share the rest between the equal sides. A square of perimeter \(30\) cm has sides of \(7.5\) cm.