Trapezoid Area Calculator

Formula

A = ((B + b) × h) / 2

Area of a trapezium

A trapezium (US: trapezoid) has two parallel sides, the major base B and the minor base b, plus the perpendicular distance between them, which is the height h. Its area is A = (B + b)·h / 2. Take B = 8 m, b = 4 m and h = 5 m and you get A = (8 + 4)·5 / 2 = 30 m². You can also write it as A = M·h, where M = (B + b)/2 is the midsegment, or median.

There are three kinds. A right trapezium has one non-parallel side perpendicular to the bases, so two of its angles are right angles. An isosceles one has non-parallel sides of equal length and an axis of symmetry. A scalene one has all four sides different. Whichever kind you have, measure the height straight across between the bases, never along a slanted side.

Applications

This same formula is what makes the trapezoidal rule work for numerical integration: ∫f(x)dx ≈ Σ((f(x_i) + f(x_{i+1}))/2)·Δx, which shows up all over engineering software. Surveyors lean on it too, since an irregular plot is easy to chop into trapezia and add up. And you find it in the geometry of sloped roofs, dam cross-sections and stadium seating, anywhere two horizontal levels meet through inclined sides.

FAQ

What height do I use? The perpendicular distance between the two parallel bases, not the length of a slanted side. If all you have is the slanted side and an angle, work the height out with h = side · sen(angle).

What is the midsegment? It's the segment that joins the midpoints of the two non-parallel sides. Its length comes out to the average of the bases, M = (B + b)/2, and that's exactly why A = M·h gives the same answer.

Is a parallelogram a trapezium? It depends on the definition you use. Under the inclusive one (any quadrilateral with at least one pair of parallel sides), yes. Under the exclusive one (exactly one pair), no. The convention differs from country to country.

How accurate is the trapezoidal rule? The error shrinks with Δx², so cutting the step size in half drops the error to roughly a quarter of what it was. When the function curves a lot, Simpson's rule, which fits parabolic arcs instead of straight lines, gets there faster.