Geometry & Measures
Grade 2-4Perimeter & Area
Perimeter and area questions appear on every GCSE Maths paper, ranging from simple rectangle calculations to working backwards from a given area to find a missing length. This lesson covers finding the perimeter of rectangles, triangles and irregular shapes, the area of rectangles, triangles and parallelograms, and rearranging these formulae to find a missing side.
What you need to know
- Perimeter is the distance all the way round; area is the space inside, in square units.
- Rectangle area is length times width; triangle area is one half times base times perpendicular height.
- Parallelogram area is base times perpendicular height; trapezium is one half of (a plus b) times height.
- Keep units consistent and give area in square units.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding perimeter
Perimeter is the total distance around the outside of a shape, found by adding up the lengths of all its sides. For a rectangle, this can be written as 2 × (length + width), since opposite sides are equal.
A rectangle with length 8 cm and width 5 cm has a perimeter of 2 × (8 + 5) = 26 cm. An irregular shape with sides of 3 cm, 4 cm, 5 cm, 6 cm and 7 cm has a perimeter of 3 + 4 + 5 + 6 + 7 = 25 cm, found simply by adding every side.
Area of rectangles and triangles
The area of a rectangle is length × width. The area of a triangle is ½ × base × perpendicular height, where the perpendicular height is measured at a right angle to the base, not along a sloping side.
A rectangle measuring 8 cm by 5 cm has an area of 8 × 5 = 40 cm². A triangle with a base of 8 cm and a perpendicular height of 5 cm has an area of ½ × 8 × 5 = 20 cm².
Area of a parallelogram
The area of a parallelogram is base × perpendicular height, using the same idea as a rectangle since a parallelogram can be rearranged into a rectangle of the same base and height.
A parallelogram with a base of 9 cm and a perpendicular height of 4 cm has an area of 9 × 4 = 36 cm².
Working backwards from perimeter or area
When a question gives the perimeter or area and asks for a missing length, set up an equation using the correct formula, then solve it for the unknown.
A rectangle has a perimeter of 30 cm and a length of 9 cm. Since 2 × (9 + w) = 30, then 9 + w = 15, so the width is 6 cm. A rectangle has an area of 48 cm² and a width of 6 cm, so its length is 48 ÷ 6 = 8 cm.
Worked examples
Three exam-style questions, fully solved
Find the perimeter of a rectangle with length 8 cm and width 5 cm.
Easy- 1.Add the length and width, then double the result: 2 × (8 + 5)
Answer: 26 cm
A rectangle has a perimeter of 30 cm. Its length is 9 cm. Find its width.
Medium- 1.Set up an equation using the perimeter formula: 2 × (9 + w) = 30
- 2.Divide both sides by 2: 9 + w = 15
- 3.Subtract 9 from both sides: w = 15 - 9
Answer: w = 6 cm
Rectangle A measures 6 cm by 8 cm. Rectangle B measures 5 cm by 10 cm. Which rectangle has the larger area, and by how much?
Hard- 1.Find the area of rectangle A: 6 × 8 = 48 cm²
- 2.Find the area of rectangle B: 5 × 10 = 50 cm²
- 3.Compare the two areas: 50 - 48
Answer: Rectangle B is larger, by 2 cm²
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A rectangle is 9 cm long and 4 cm wide. Work out its area.
A rectangle measures 12 cm by 5 cm. Work out its perimeter.
A triangle has a base of 10 cm and a perpendicular height of 6 cm. Work out its area.
A parallelogram has a base of 8 cm and a perpendicular height of 5 cm. Work out its area.
A trapezium has parallel sides of 6 cm and 10 cm, and a perpendicular height of 4 cm. Work out its area.
A square has an area of 49 cm². Work out its perimeter.
Avoid these
The mistakes students make most often
Using a sloping side instead of the perpendicular height when finding the area of a triangle or parallelogram.
Forgetting to halve the result when finding the area of a triangle, giving the area of a rectangle with the same base and height instead.
Adding only two sides of a rectangle instead of all four when finding perimeter, or forgetting to double length plus width.
Mixing up the formulae for perimeter and area, for example multiplying length by width when asked for perimeter.
Forgetting the units for area (cm²) and writing the same units as perimeter (cm) instead.
FAQ
Questions students and parents ask
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