Trapezoid Area Calculator

A trapezoid (a trapezium in UK English) has exactly one pair of parallel sides. Those two are the bases, usually labelled a for the shorter and b for the longer. The perpendicular distance between them is the height h.

A = ½ × (a + b) × h

Read aloud: half the sum of the parallel sides, times the perpendicular height. The (a + b)/2 term is the average of the two bases, and it is also the length of the mid-segment, the line drawn halfway between them.

The word perpendicular in that sentence is doing real work. The height is the straight-across distance between the two parallel lines, not the length of a sloped leg. Measuring the leg instead is the single most common way this calculation goes wrong, and on a shallow trapezoid the leg can be half again as long as the height.

Where trapezoids appear in real measurements:

• Gable end of a house with a kneewall. A two-story house with a partial roof slope creates trapezoidal wall sections.
• Retaining wall cross-section. Almost every retaining wall is wider at the base than at the top, which makes the section a trapezoid. That area times the wall length gives the concrete volume.
• Land surveys. Irregular plots get split into trapezoids to be measured, and the Trapezoidal Rule in calculus is the same idea applied to a curve.
• Stage and concert venues. Auditoriums that fan outward from the stage have trapezoidal floor plans, which give better sightlines than a rectangle of the same area.
• Garden bed planters with sloped sides, usually a wider bottom for stability.
• Roof gambrel sections on a barn-style roof. The lower, steeper section is a trapezoid.

Worked example: retaining wall cross-section

You’re pouring a concrete retaining wall 50 ft long. The cross-section is a trapezoid: 24 in wide at the base, 12 in wide at the top, and 4 ft (48 in) tall. Cross-section area = 0.5 × (12 + 24) × 48 = 864 sq in = 6 sq ft. Total concrete volume = 6 × 50 = 300 cubic feet = 11.1 cubic yards. Concrete is sold by the cubic yard, so order 12 to have a slight margin.

The “mid-segment shortcut”:

If you only know the mid-segment m, the length of a line drawn halfway between the two parallel sides, then A = m × h. Surveyors often measure that directly because reaching both bases can mean crossing the plot twice.

Two shortcuts that do NOT work

Averaging the two bases and squaring it. Multiplying the longer base by the height and taking off a bit. Both come up on site and both are wrong, sometimes by a lot: for a = 2, b = 10, h = 4 the real area is 24, while “longer base times height minus a bit” starts from 40 and needs a 40% haircut nobody guesses correctly.

Sanity check: if a = b the shape is a parallelogram and the formula reduces to a × h. If a = 0 it is a triangle and the formula gives ½ × b × h. One formula covers the rectangle, the parallelogram, the trapezoid and the triangle, which is why it is worth learning this one rather than four.