Geometry · GCSE Maths
Area of 2D shapes
GCSE Maths area: rectangles, triangles, parallelograms, trapeziums and circles (πr²), plus compound shapes split into parts, with units in cm² and m² and conversions between them.
Write the formula, substitute with units, then add or subtract areas for compound shapes. Never mix cm and m without converting first.
The important bits
What you need to know
- 1
Rectangle: A = l × w. Parallelogram: A = b × h, where h is the perpendicular height to the base, not the slanted side.
- 2
Triangle: A = ½ × b × h. The height must meet the base at a right angle. If you only have two sides and the included angle, A = ½ab sin C.
- 3
Trapezium: A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular distance between them.
- 4
Circle: A = πr². Diameter d means r = d/2. On a calculator paper use π or the π button; on non-calculator leave π in the answer unless told otherwise.
- 5
Compound shapes: split into rectangles, triangles or semicircles, find each area, then add. For a shape with a bite taken out, subtract the missing piece.
- 6
Units: area is always square units — cm², m², mm². 1 m² = 10 000 cm² because 1 m = 100 cm and 100 × 100 = 10 000.
- 7
Show every step: formula, substitution, calculation, units. A number without cm² loses the final mark on many papers.
- 8
Check reasonableness: a triangle with base 6 cm and height 4 cm has area 12 cm², not 24 cm² — the ½ matters.
Quotations worth analysing
Short evidence. Real method.
“A = πr²”
Radius, not diameter. If the question gives 10 cm across, halve it before squaring. Squaring the diameter and multiplying by π is a common full-mark loss.
“A = ½(a + b)h”
h is the vertical (perpendicular) gap between the parallels, not the length of a slanted leg. Mark h on the diagram before you substitute.
“Split, find each area, then add”
Draw dotted lines to show how you split the shape. Examiners award method marks for a clear partition even if one arithmetic slip follows.
Go deeper
Perpendicular height is not the slanted side
A parallelogram with base 8 cm and slanted side 5 cm does not have area 40 cm² unless the perpendicular height is 5 cm. Drop a perpendicular from the top to the base and label that height h. If h = 4 cm, A = 8 × 4 = 32 cm². The same rule applies to triangles: the height must meet the base at 90°. In a right-angled triangle, the two legs are base and height. In an obtuse triangle, the height may fall outside the base — extend the base line if you need to. For trapeziums, only the parallel sides go into (a + b). The non-parallel sides are not substituted into the formula. Sketch, mark h with a right-angle symbol, then write A = ½(a + b)h with numbers.
Go deeper
Circles and semicircles in compound shapes
A lawn that is a rectangle with a semicircular end is rectangle plus semicircle, not a full circle. Semicircle area is ½πr². If the semicircle sits on a side of length 12 m, that side is the diameter, so r = 6 m. Rectangle 12 × 8 = 96 m²; semicircle ½ × π × 6² = 18π m²; total 96 + 18π m², or 152.5 m² to 1 d.p. if π ≈ 3.14. Compound questions often give a diagram in one unit and ask for m² — convert each length before area, not after. Subtracting a circular hole from a square plate: square minus full circle, with the hole’s radius clear on the diagram.
Go deeper
Converting between cm² and m²
Linear conversion: 1 m = 100 cm. Area conversion: multiply the linear factor twice. 1 m² = 100 cm × 100 cm = 10 000 cm². To change 3.5 m² to cm², multiply by 10 000 → 35 000 cm². To change 250 cm² to m², divide by 10 000 → 0.025 m². A rectangle 200 cm by 150 cm has area 30 000 cm² = 3 m². Converting only one dimension and then multiplying is a classic error: 2 m by 50 cm is not 100 m²; convert 50 cm to 0.5 m first, then 2 × 0.5 = 1 m². Write the unit conversion as a separate line so the examiner sees you know area is squared.
See the idea in action
Find the area of a trapezium with parallel sides 5 cm and 9 cm, and perpendicular height 6 cm. Step 1: Write the formula: A = ½(a + b)h. Step 2: Identify a = 5 cm, b = 9 cm, h = 6 cm (perpendicular between the parallels). Step 3: A = ½(5 + 9) × 6 = ½ × 14 × 6. Step 4: A = 7 × 6 = 42 cm². Check: average of the parallel sides is 7 cm; times height 6 cm gives 42 cm², sensible for a medium trapezium.
Exam technique
Turn knowledge into marks
Label the formula before substituting. For compound shapes, draw how you split the shape and write “Rectangle + triangle” or similar before adding areas.
Common mistakes
Do not give these marks away
- 01
Using the slanted side instead of the perpendicular height in a triangle, parallelogram or trapezium.
- 02
Substituting the diameter into πr² without halving it first, or using πd².
- 03
Adding areas in mixed units (cm and m) without converting, or forgetting the ½ in triangle and trapezium formulas.
A circle has radius 5 cm. What is its area?
A25π cm²
B10π cm²
C5π cm²
Dπ cm²
Show the answer
25π cm². A = πr² = π × 5² = 25π cm². 10π cm² would be π × 5 × 2, confusing circumference with area. 5π cm² forgets to square the radius.
Quick questions
If this is the bit you searched
When do I use ½ × base × height for a triangle?
Whenever you know the base and the perpendicular height. If you know two sides and the included angle, use A = ½ab sin C instead.
How do I find the area of a compound shape?
Split it into rectangles, triangles, circles or semicircles. Find each area, then add. If part is removed, subtract that area from the larger shape.
How do I convert m² to cm²?
Multiply by 10 000, because 1 m = 100 cm and area squares that factor: 100 × 100 = 10 000.
Should I use π or 3.14?
Use the π button on calculator papers unless the question specifies 3.14. On non-calculator papers, leave answers in terms of π unless told otherwise.