This note covers how to find the perimeter and area of rectangles, triangles, parallelograms, trapezia and compound shapes made from them, and how to convert between units of length and area. Perimeter and area calculations are used in building, design, farming and costing, so accuracy with units matters.
Key ideas
The perimeter of a shape is the total distance around its edge. It is measured in units of length such as cm or m. The area of a shape is the amount of flat surface it covers. It is measured in square units such as cm\(^2\) or m\(^2\).
- Rectangle: area is length \(\times\) width. Perimeter is \(2 \times (\text{length} + \text{width})\).
- Triangle: area is half the base times the perpendicular height. The height must be at right angles to the base, not the sloping side.
- Parallelogram: area is base times perpendicular height. Opposite sides are equal and parallel.
- Trapezium: area is half the sum of the two parallel sides, multiplied by the perpendicular distance between them.
Compound shapes are made from simple shapes joined together or with pieces cut out. There are two approaches. You can split the shape into rectangles and triangles, find each area, and add. Or you can enclose the shape in a larger rectangle and subtract the missing parts. Choose whichever needs fewer calculations.
For the perimeter of a compound shape, add up the lengths of the outside edges only. Some edge lengths are not given, so you may need to work them out from other lengths. If a rectangle has a corner cut out in a right-angled step, the perimeter stays the same as the perimeter of the original rectangle.
Working with units. Convert all lengths to the same unit before you calculate. Be careful when converting areas. Since \(1\ \text{m} = 100\ \text{cm}\), we have \(1\ \text{m}^2 = 100 \times 100 = 10\,000\ \text{cm}^2\). Similarly \(1\ \text{cm}^2 = 100\ \text{mm}^2\) and \(1\ \text{km}^2 = 1\,000\,000\ \text{m}^2\).
Working backwards. Many questions give the area and ask for a missing length. Substitute the known numbers into the area formula, then rearrange. For example, if a triangle has area 30 cm\(^2\) and base 12 cm, then \(30 = \tfrac{1}{2} \times 12 \times h\), so \(h = 5\) cm.
Answers should have units. Give lengths in cm or m, and areas in cm\(^2\) or m\(^2\). If an answer is not exact, round to three significant figures unless told otherwise.
Real measurements are rarely tidy. If the question gives a length in centimetres and another in metres, choose one unit for the whole calculation. Converting at the start is safer than converting the final answer, because an area conversion uses a squared factor.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Rectangle \(A = lw\) | \(l\) is length and \(w\) is width. Perimeter \(P = 2(l + w)\). |
| Triangle \(A = \tfrac{1}{2}bh\) | \(b\) is the base and \(h\) is the perpendicular height. |
| Parallelogram \(A = bh\) | \(h\) is the perpendicular height, not the slant side. |
| Trapezium \(A = \tfrac{1}{2}(a + b)h\) | \(a\) and \(b\) are the parallel sides. \(h\) is the distance between them. |
| \(1\ \text{m}^2\) | \(10\,000\ \text{cm}^2\). |
| \(1\ \text{cm}^2\) | \(100\ \text{mm}^2\). |
| Compound shape | Split into simple shapes and add, or subtract cut-out parts from an enclosing shape. |
Worked examples
Example 1 A flower bed is a trapezium. Its parallel sides are 7 m and 11 m and its area is 54 m\(^2\). Find the distance between the parallel sides.
Step 1: write the formula. \(A = \tfrac{1}{2}(a + b)h\).
Step 2: substitute. \(54 = \tfrac{1}{2}(7 + 11)h = 9h\).
Step 3: divide by 9. \(h = 6\).
Step 4: check. \(\tfrac{1}{2} \times 18 \times 6 = 54\).
Answer: 6 m
Example 2 An L-shaped patio is made from a 10 m by 8 m rectangle by removing a 4 m by 3 m rectangle from one corner. Find the area and the perimeter of the patio.
Step 1: area of the large rectangle \(= 10 \times 8 = 80\ \text{m}^2\).
Step 2: area removed \(= 4 \times 3 = 12\ \text{m}^2\).
Step 3: area of the patio \(= 80 - 12 = 68\ \text{m}^2\).
Step 4: the cut-out is a right-angled step in one corner. The new edges run 4 m across and 3 m down, and replace 4 m and 3 m of the original edges. So the perimeter equals that of the original rectangle: \(2(10 + 8) = 36\) m.
Answer: area 68 m\(^2\); perimeter 36 m.
Example 3 A wall is 4.5 m wide and 2.8 m high. It has a window 1.2 m wide and 1.5 m high. One litre of paint covers 9 m\(^2\), and tins hold 1 litre each at £14.50 per tin. Find the cost of one coat of paint on the wall, not including the window.
Step 1: area of the wall \(= 4.5 \times 2.8 = 12.6\ \text{m}^2\).
Step 2: area of the window \(= 1.2 \times 1.5 = 1.8\ \text{m}^2\).
Step 3: area to paint \(= 12.6 - 1.8 = 10.8\ \text{m}^2\).
Step 4: litres needed \(= 10.8 \div 9 = 1.2\). You cannot buy part of a tin, so 2 tins are needed.
Step 5: cost \(= 2 \times 14.50 = 29\).
Answer: £29.00
Common errors
- Using the sloping side instead of the perpendicular height in a triangle, parallelogram or trapezium.
- Forgetting the half in the triangle and trapezium formulae.
- Converting areas with the length factor. \(1\ \text{m}^2\) is \(10\,000\ \text{cm}^2\), not \(100\ \text{cm}^2\).
- Including internal edges when finding the perimeter of a compound shape.
- Leaving out units, or giving square units for a perimeter.
Exam tip: when a shape is compound, draw the dividing lines on the diagram and label the lengths you have worked out. In cost problems, read the last line carefully: you may need to round up to a whole number of tins or packs.
Check yourself
- Find the area of a triangle with base 14 cm and perpendicular height 9 cm.
Answer
\(\tfrac{1}{2} \times 14 \times 9 = 63\ \text{cm}^2\). - A parallelogram has base 8.5 cm and perpendicular height 4 cm. Find its area.
Answer
\(8.5 \times 4 = 34\ \text{cm}^2\). - Convert 2.5 m\(^2\) to cm\(^2\).
Answer
\(2.5 \times 10\,000 = 25\,000\ \text{cm}^2\). - A square of side 6 cm has an isosceles triangle of base 6 cm and perpendicular height 4 cm fixed on top, along one side. Find the total area.
Answer
Square: \(6 \times 6 = 36\). Triangle: \(\tfrac{1}{2} \times 6 \times 4 = 12\). Total \(= 48\ \text{cm}^2\).