Area of a trapezium

In a trapezium only one of the pairs of sides is parallel. The angle sum is still 360, as in all other quadrilaterals. The area formula for a trapezium can look a bit more complicated than the others, but it actually isn't, and this guide takes it one step at a time.

When do I use this?

Whenever the shape has exactly one pair of parallel sides. If both pairs are parallel you're looking at a rectangle or a parallelogram, and those have their own guides: area of a square and a rectangle and area of a parallelogram.

The formula

Here $x$ and $y$ are the parallel sides, and $h$ is the height, the dashed line that stands at a right angle between them.

$$A = \frac{1}{2} \times h \times (x + y)$$

We add the two parallel sides together and multiply by the height, and after that we multiply by a half.

The procedure

1. Find the two parallel sides, $x$ and $y$, and add them together.
2. Multiply by the height $h$.
3. Multiply by a half.

The bracket around $x + y$ matters. The two parallel sides are added first, and only then does the height come in.

Worked example

If a trapezium has the parallel side lengths 2 and 4, with a height of 6, the area is

$$A = \frac{1}{2} \times 6 \times (2 + 4) = 18$$

Step by step: the parallel sides add up to $2 + 4 = 6$. Times the height gives $6 \times 6 = 36$. Half of that is $18$.

Common mistakes

• "The formula is too complicated to remember." It really isn't. Add the two parallel sides, multiply by the height, take half. Three moves.
• "I add the slanted sides." The two sides you add are the parallel ones, $x$ and $y$ in the figure. The slanted sides aren't in the formula at all.
• "The height is one of the slanted sides." The height is the dashed line that stands straight between the two parallel sides, at a right angle to them.

Related

The trapezium is the last of the four shapes in Quadrilaterals: square, rectangle, parallelogram and trapezium. Its formula sits between the two simpler ones: the rectangle, where you multiply two sides, and the parallelogram, where you multiply the base by the height.