Trapezoid Area Calculator

A trapezoid (called a trapezium in UK English) has exactly one pair of parallel sides. The two parallel sides are the “bases” — usually labeled a (shorter) and b (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 length of the parallel sides — which is also the length of the mid-segment, the line halfway between the two bases.

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. Most retaining walls are wider at the base than at the top — a trapezoidal cross-section. The area of that cross-section, times the wall length, gives concrete volume.
• Land surveys. Irregular plots of land can often be split into trapezoids for area calculation. The Trapezoidal Rule in calculus comes from this idea.
• Stage and concert venues. Many auditoriums fan outward from a stage — trapezoidal floor plans give better sightlines than rectangles.
• Garden bed planters with sloped sides (often used for raised beds with a wider bottom for stability).
• Roof gambrel sections on a barn-style roof. The lower steep section is trapezoidal.

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. Some surveyors measure this directly because it’s easier than getting both bases.

A right trapezoid (two right angles, one slanted side) is common in architecture — picture a doorway with one vertical jamb and one sloped jamb. Even with the slant, the area formula is the same.

Sanity check: if a = b, the shape is a parallelogram and the formula reduces to a × h (just the regular rectangle/parallelogram formula). If a = 0, the shape is a triangle and the formula gives ½ × b × h. The trapezoid formula gracefully covers both edge cases.