Perimeter is the total distance around a shape, measured in units such as cm. Area is the space inside, measured in square units such as cm². The key area formulae are length × width for a rectangle, ½ × base × height for a triangle, base × height for a parallelogram and ½(a + b)h for a trapezium. Compound shapes are split into simpler shapes.
Area formulae
- Rectangle: length × width
- Triangle: ½ × base × perpendicular height
- Parallelogram: base × perpendicular height
- Trapezium: ½(a + b)h, where a and b are the parallel sides
- Perimeter: add all the outside edges
Perimeter and area
Perimeter is a length, the sum of all the outside edges, so its units are cm or m. Area measures the surface inside and uses square units, cm² or m². A rectangle 9 cm by 5 cm has perimeter 9 + 5 + 9 + 5 = 28 cm and area 9 × 5 = 45 cm².
Triangles and parallelograms
The area of a triangle is ½ × base × height, where the height is measured at right angles to the base, not along a sloping side. A parallelogram is base × perpendicular height, because it can be rearranged into a rectangle.
Trapeziums
A trapezium has one pair of parallel sides, a and b. Its area is ½(a + b)h: add the parallel sides, halve, then multiply by the perpendicular distance between them. For parallel sides 7 cm and 11 cm, 4 cm apart, the area is ½ × 18 × 4 = 36 cm².
Compound shapes
Split a compound shape into rectangles, triangles or trapeziums, find each area and add them. Sometimes it is easier to subtract: a 10 cm by 6 cm rectangle with a 4 cm by 3 cm corner removed has area 60 − 12 = 48 cm². For the perimeter, work out any missing side lengths from the ones given, and add every outside edge.
Finding a missing length
If the area is known, rearrange the formula. A triangle with area 45 cm² and height 9 cm has base 2 × 45 ÷ 9 = 10 cm. A rectangle with area 60 cm² and length 12 cm has width 5 cm.
Worked examples
Example 1. A rectangle is 9 cm by 5 cm. Work out its perimeter and area.
- Perimeter: 9 + 5 + 9 + 5.
- Area: 9 × 5.
Answer: Perimeter 28 cm; area 45 cm².
Example 2. A triangle has base 13 cm and perpendicular height 8 cm. Work out its area.
- ½ × 13 × 8.
Answer: 52 cm².
Example 3. A trapezium has parallel sides 7 cm and 11 cm and height 4 cm. Work out its area.
- ½ × (7 + 11) × 4 = ½ × 18 × 4.
Answer: 36 cm².
Example 4. A 10 cm by 6 cm rectangle has a 4 cm by 3 cm rectangle cut from one corner. Work out the area of the remaining shape.
- Large rectangle: 10 × 6 = 60.
- Cut-out: 4 × 3 = 12.
- 60 − 12.
Answer: 48 cm².
Example 5. A triangle has area 45 cm² and perpendicular height 9 cm. Work out its base.
- ½ × base × 9 = 45, so base × 9 = 90.
Answer: Base = 10 cm.
Common mistakes
- Using the sloping side as the height. The height must be perpendicular to the base.
- Forgetting the ½ for a triangle. Area of a triangle = ½ × base × height.
- Using the wrong sides in a trapezium. a and b are the two parallel sides.
- Wrong units. Perimeter is in cm; area is in cm².
- Missing edges in a compound perimeter. Work out every unlabelled side before adding.
How area and perimeter are examined
Area and perimeter are core Foundation topics on Edexcel papers, often hidden inside problem-solving questions about cost or materials. Questions ask you to:
- find the area and perimeter of rectangles, triangles and trapeziums
- find the area of a compound shape
- find a missing length from a given area
- solve a context problem, such as the cost of carpeting a room
The area and perimeter worksheet includes answers, and the GCSE geometry revision engine generates more questions.
Next, see circles and volume and surface area.
Frequently asked questions
How do area and perimeter differ?
Perimeter is the distance around a shape; area is the space inside it.
What is the formula for the area of a trapezium?
½(a + b)h, where a and b are the parallel sides and h is the height.
How do you find the area of a compound shape?
Split it into simpler shapes, find each area, and add them (or subtract a missing piece).