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.
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
Avoid These
The most common mistakes students make.
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 parents and students ask.
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