Area of a Trapezium, the Formula and the Questions That Hide It

A trapezium is a four-sided shape with exactly one pair of parallel sides. Its area is

Area = (1/2) x (a + b) x h,

where a and b are the two parallel sides and h is the perpendicular distance between them. The formula is short, but two things about it are worth understanding, because that is where the marks actually move.

Why the formula works

Take the average of the two parallel sides, (a + b)/2. That is the width the trapezium would have if it were a rectangle of the same height. Multiply that average width by the height and you have the area. So a trapezium is just a rectangle whose width is the average of its two parallel sides. Seeing it this way means you never have to memorise the formula as a string of symbols; you can rebuild it.

A worked check: parallel sides 8 cm and 5 cm, height 4 cm. Area = (1/2)(8 + 5)(4) = (1/2)(13)(4) = 26 cm2. The average width is 6.5 cm, and 6.5 x 4 = 26, the same answer, which is the formula and the reasoning agreeing.

Where the height is measured

The single most common error is using a slanted side as the height. The h in the formula is the perpendicular distance between the two parallel sides, not the length of the sloping edge. If a question gives you a slant length and an angle, you often have to find the true perpendicular height first, with trigonometry or Pythagoras, before the area formula applies. Reading which length is perpendicular is half the question.

When the trapezium is hidden

In Paper 2, the trapezium is rarely handed to you cleanly. It is the leftover when a triangle is cut from a rectangle, or one face of a prism, or the cross-section of a channel. The skill GPA teaches is to identify the parallel sides and the perpendicular height inside the larger figure first, label them, and only then apply the formula. A second worked case: parallel sides 12 cm and 7 cm, height 6 cm gives (1/2)(12 + 7)(6) = 57 cm2. The arithmetic is easy once the three correct lengths are picked out; picking them out is the real task.

The teaching point

The trapezium area is a rectangle in disguise, average the parallel sides, times the height, and the height is always the perpendicular one. Most lost marks here are not arithmetic; they are using the wrong length as the height, or failing to spot the trapezium inside a composite figure. Naming the three lengths before substituting is the habit that fixes both.

This figure-reading discipline runs from PSLE area and perimeter right up into O-Level mensuration. If your child knows the formula but mislabels the height or misses the shape inside a diagram, GPA's programmes work on exactly that reading skill. A short diagnostic consult will show where the labelling slips.

Both worked areas on this page were checked independently before publishing.