Trapezoid area:
average the parallel sides, multiply by the height

What counts as a trapezoid

A trapezoid (called a trapezium in British usage) is a four-sided shape with one pair of parallel sides. Those parallel sides are the bases, and the distance between them, measured at a right angle, is the height. The other two sides, the legs, may slope inwards, outwards or even differ in length. Roads in cross-section, buckets, hoppers, drainage channels and the faces of many roofs all take this shape.

Area is still the amount of flat surface inside the outline, which you can think of as the number of unit squares that fit in it.

The formula

The logic is intuitive. A trapezoid is stretched evenly from the short side to the long side, so on average it is as wide as the mean of the two. Multiply that average width by the height and you have the area.

Worked example: parallel sides 6 and 10, height 5

A trapezoid with these dimensions could be a garden bed 6 m wide at the top, 10 m at the bottom, and 5 m deep, or a diagram in centimetres; the unit simply becomes the square of whatever you used.

Checking by splitting the shape

There is a second route to the same number, which makes a good verification. Cut off the two triangular ends and what is left in the middle is a rectangle as wide as the shorter base. Here that rectangle is 6 × 5 = 30. The leftover width is 10 − 6 = 4, shared between the two ends. Placed together they form triangles with a total base of 4 and height 5, so their area is 4 × 5 ÷ 2 = 10. Adding gives 30 + 10 = 40, matching the formula.

This works for any trapezoid, even when the two sloping ends are different sizes, because only their combined width matters.

Real example: a drainage channel cross-section

A trapezoidal channel is 3.0 m wide at the top, 1.2 m wide at the base and 0.8 m deep. Cross-sectional area is (3.0 + 1.2) ÷ 2 × 0.8 = 2.1 × 0.8 = 1.68 m².

Multiply the cross-section by a length to estimate volume: a 50 m stretch of this channel involves 1.68 × 50 = 84 m³ of excavation, ignoring any lining thickness.

Why the average-width trick works

Imagine the trapezoid duplicated, rotated through 180 degrees, and joined to the original along a leg. The two pieces make a parallelogram whose base is a + b and whose height is h. Its area is (a + b) × h, and the trapezoid is half of that. Dividing by two is therefore the same as averaging the two bases.

The picture explains why the shape of the legs does not matter. Whether the sides lean left, right or both ways, the combined width of the two bases is what counts. It also explains why a rectangle is a special case: when a equals b the average is just a, and you get the base times the height you already know. A triangle is another special case, with one base shrunk to zero, which gives ½ × b × h.

Applications in building and surveying

Cross-sections are the most common practical use. Roads, embankments, canals, cuttings and foundations are often drawn as trapezoids, and the area multiplied by length gives volume. Earthwork estimates frequently use the average end area method: take the cross-section area at two stations, average them, and multiply by the distance between stations.

Two stations 20 m apart with cross-sections of 1.68 m² and 2.10 m² have an average of 1.89 m², giving a volume of 1.89 × 20 = 37.8 m³. Sloped hopper walls, tapered boards and some roof faces also take this form, and trapezoid panels are cut from sheet by calculating their area to estimate waste. For unusual shapes, break them into a trapezoid plus triangles and sum.

Remember to keep every length in one unit, then convert the final area or volume at the end.

To practise, try another case: a retaining-wall elevation with a top length of 12 m, a base length of 18 m and a height of 3 m. The area is (12 + 18) ÷ 2 × 3 = 45 m². If the wall is plastered on one face at a rate quoted per square metre, that is the quantity to price. A sanity check: the area must lie between the smaller rectangle 12 × 3 = 36 and the larger 18 × 3 = 54, and 45 is exactly midway because the average of the bases is 15.

Bounding like this is a quick way to catch slips before you trust a result.

Mistakes to avoid

• Using the slanted leg as the height. The height must be perpendicular to the parallel sides.
• Averaging the legs instead of the parallel sides.
• Mixing units, for example one base in centimetres and the other in metres.
• Treating any quadrilateral as a trapezoid. If neither pair of sides is parallel the formula does not apply.

When the shape is irregular, break it into a rectangle tool and a triangle tool, as above, and add the parts.