Visualgebra

Area of a trapezium

Two triangles, one diagonal, and a height both of them happen to share.

A parallelogram’s two parallel sides always match. A trapezium’s do not have to.

A trapezium, A, B, C and D. AB and DC are always parallel, however the figure is dragged, but they do not have to be the same length.

A trapezium, A, B, C and D, where A, B and D are draggable and C is free to slide along the line through D parallel to AB, so the two parallel sides can be different lengths. The perpendicular height from D to line AB is drawn, meeting it at F. A live reading gives both parallel sides, the height, and the trapezium area formula. This needs JavaScript switched on.

Drag any point. (AB + DC) ÷ 2 × height always gives the trapezium's area.

Try to make AB much longer than DC, or the reverse — the two parallel sides stretch independently.

A quadrilateral with exactly one pair of parallel sides is a trapezium — named back when the parallelogram family tree branched out to it, measured for the first time here. AB and DC are its two parallel sides; the height is the perpendicular distance between the lines they sit on, same meaning as every area lesson so far.

Two triangles sharing one height

Draw diagonal AC. It splits the trapezium into two triangles, ABC and ACD. Triangle ABC has base AB; its height, measured from C perpendicular to line AB, is exactly the distance between the two parallel lines — the same height DF this widget already draws, since C sits on the line through D parallel to AB. Triangle ACD has base DC; its height, measured from A perpendicular to line DC, is that same distance again, for the same reason in reverse.

A△ABC=AB×h2,A△ACD=DC×h2A_{\triangle ABC} = \frac{AB \times h}{2}, \qquad A_{\triangle ACD} = \frac{DC \times h}{2}

Add them:

Atrapezium=AB×h2+DC×h2=(AB+DC)×h2A_{\text{trapezium}} = \frac{AB \times h}{2} + \frac{DC \times h}{2} = \frac{(AB + DC) \times h}{2}

Congruence was never actually required here — only a shared height, and a shared height doesn’t demand equal bases. That’s the whole difference from a parallelogram: same trick, one fewer assumption.

One to watch out for.

A parallelogram is the special case where AB and DC happen to be equal. Put AB = DC into this formula and it becomes AB × h — the parallelogram formula, exactly, falling out rather than being a separate fact.

A trapezium with parallel sides 6 and 10 units, and height 5 units.

(6+10)×52=40 square units\frac{(6 + 10) \times 5}{2} = 40 \text{ square units}

A trapezium with parallel sides 3 and 9 units, and height 4 units.

(3+9)×42=24 square units\frac{(3 + 9) \times 4}{2} = 24 \text{ square units}

Same rule as every shape in this chapter: only the base and the perpendicular height enter the formula.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.