How To Calculate The Area Of A Trapezoid

A trapezoid, called a trapezium in British English, is a quadrilateral with exactly one pair of parallel sides. Those two sides are usually called the bases even though they are not necessarily horizontal, and the other two sides are the legs. The shape covers everything from a garden bed that tapers at one end to the cross-section of a road embankment.

The area formula is the average of the two parallel sides multiplied by the height. Adding them and halving gives a length called the median, which is the width of the trapezoid exactly halfway between the two bases. The area is then simply the median times the height, which is the same arithmetic a rectangle would use.

That reading of the formula is worth holding on to, because it explains why the answer sits between the area of the small rectangle and the area of the large one. A trapezoid with parallel sides of 6 and 10 and a height of 4 occupies the same space as a rectangle 8 wide and 4 tall, and 8 is exactly halfway between 6 and 10.

The height is the perpendicular distance between the parallel sides, not the length of a slanted leg. This distinction matters more for a trapezoid than for almost any other shape, because the legs are usually visibly longer than the perpendicular gap. Using a leg length in place of the height overstates the area by the ratio of the leg to the perpendicular.

Degenerate cases show the formula is a generalisation rather than a special rule. If the two parallel sides are equal the trapezoid becomes a rectangle, and the average of two equal numbers is that number, so the formula collapses to width times height. If one parallel side shrinks to zero the shape becomes a triangle, and the formula becomes half the base times the height.

An isosceles trapezoid has legs of equal length and mirror symmetry about the vertical centre line. That symmetry makes the leg length computable from the parallel sides and the height: the horizontal offset of each leg is half the difference between the bases, and the leg is the hypotenuse of a right triangle with that offset and the height as its two shorter sides.

Perimeter is a separate calculation and needs the leg lengths. For an isosceles trapezoid the legs are equal, so the perimeter is the two parallel sides plus twice one leg. For a scalene trapezoid each leg has to be measured or derived separately, and the area formula alone gives no information about them.

Trapezoids appear wherever a quantity changes linearly between two ends. The area under a straight line that starts at one height and ends at another is a trapezoid, which is why the trapezoid rule for numerical integration uses this exact formula to approximate the area under a curve.

In construction the trapezoid shows up as tapered sites, sloped driveways, embankments and the cross-section of a trench. The arithmetic is the same in every case, but the height is almost always the dimension that has to be measured carefully, since it is the perpendicular gap rather than the sloping face.

One practical shortcut: if you know the median directly, the formula is just median times height. Surveying a long strip, it is often easier to measure the width at the midpoint than to measure both ends and average them, and the result is identical.