Area of a Trapezium

How do you find the area of a shape with two parallel sides but slanted legs? Let's derive it, not memorise it. Here is a trapezium. Its two parallel sides are called the bases, and the slanted sides are the legs. Let me use clean numbers instead: bottom base 8, top base 4, and height 3. The bottom base is the longer one.

Highlight the two bases first. They must be parallel; that is the one requirement for a trapezium. The height h is the perpendicular distance between the bases, not the length of a slanted leg. Now the key idea. Make a second identical trapezium, and turn it upside down.

• bases are parallel

Rotate the copy by half a turn, then slide it against the slanted side of the first. Together they form a parallelogram. Its base is the two bases joined end to end: b one plus b two. Check the numbers: eight plus four is twelve. The top and bottom of the parallelogram are both twelve long.

A parallelogram has area base times height. So this big shape has area twelve times three, thirty six.

But that parallelogram is made of two equal trapeziums. So one trapezium is exactly half of it.

Another view: half of b one plus b two is the average of the bases. So area is average width times height. Now the worked example, one step at a time. Start by writing the formula with our numbers.

• area = mean base × height

First add the bases inside the brackets: eight plus four is twelve.

Half of twelve is six.

Six times three is eighteen square units, exactly half of the thirty six we found.

Edge case: if the top base shrinks to zero, the trapezium becomes a triangle, and the formula becomes half base times height.

If the bases are equal, it is a parallelogram, and the formula gives base times height. One formula covers all three.

Common mistake: using the slanted leg as the height. The height must be perpendicular to the bases. A longer leg gives a wrong, too large area. Real life: a swimming pool whose side view is a trapezium. A pool is 2 metres deep at one end and 1 at the other, so the side area uses these as bases. Here the parallel sides are the vertical depths, two and one, and the distance between them is ten metres, so area is half times three times ten, fifteen square metres.

• height ≠ slanted leg

Bigger idea: this is the trapezoidal rule from calculus, which approximates areas under curves by adding many thin trapeziums. Test your understanding. A trapezium has bases 6 and 10, and a slanted leg of 5, but a perpendicular height of 4. What is the area? Summary: copy, rotate, and join two trapeziums into a parallelogram, then halve it: area is half of b one plus b two, times height.