Area of Trapeziums, Rhombuses and Kites

A trapezium has exactly one pair of parallel sides — a wide end and a narrow end that never meet, joined by two slanted sides. Its area formula is a half times the sum of the parallel sides, times the height, and the neatest way to read that is as an average. Add the two parallel sides, halve them to find the average width, then multiply by the height, exactly as you would for a rectangle. For this one, 6 plus 10 is 16, half of that is 8, and 8 times 4 gives 32 square centimetres. Only the parallel sides go inside the bracket. The slanted sides never appear in the formula, and the height is the perpendicular gap between the parallel pair.

The half in the formula is not a random rule, and here is where it comes from. Take your trapezium, make a second identical copy, turn the copy upside down and slot it against the first. The narrow end of one sits beside the wide end of the other, so the two together form a parallelogram whose base is a plus b — here 10 plus 6, which is 16 — with the same height of 4. That parallelogram has an area of 16 times 4, or 64 square centimetres. But you only wanted one trapezium, so halve it: 32 square centimetres. That is exactly the formula, a half times a plus b times h, and if you can picture the pairing you can rebuild it any time you forget.

Rhombuses and kites are handled a different way again — through their diagonals, the lines joining opposite corners straight through the middle. A rhombus is a parallelogram with four equal sides, and a kite has two pairs of equal neighbouring sides. What they share is that their diagonals cross at a right angle, and that lets one formula cover both: area is a half times one diagonal times the other. A rhombus with diagonals of 8 and 6 has an area of a half times 48, which is 24 square centimetres. A kite with diagonals of 10 and 7 has an area of a half times 70, which is 35. Different shapes, same two-step calculation.

It is worth seeing where that half comes from, because then you will stop dropping it. Draw a rectangle around the rhombus so the box touches all four corners. The box is as wide as one diagonal and as tall as the other, so its area is 8 times 6, which is 48 square centimetres. Now look at the four little corner triangles that sit outside the shape but inside the box — they match the four triangles inside the shape exactly. So the rhombus fills precisely half the box, which is 24 square centimetres. A kite works the same way for the same reason: its diagonals also meet at a right angle. Multiplying the diagonals and stopping gives double the true area.

Here is the whole toolkit in one place: rectangle is length times width, triangle is a half base times height, parallelogram is base times height, trapezium is a half the sum of the parallel sides times height, and rhombus or kite is a half times the two diagonals. The habit that makes this easy is letting the measurements you are given choose the formula, rather than hunting for the shape's name. Given two diagonals, use a half x y. Given parallel sides and a height, use the trapezium formula. And when the area is given instead, run it backwards: for a trapezium of area 40 with parallel sides 6 and 10, write 40 equals a half times 16 times h, simplify to 40 equals 8h, and divide to get a height of 5 centimetres.