Trapezoid Calculator

How to use the trapezoid calculator

Match the inputs to one of four modes

The first mode uses both parallel sides and their perpendicular height to find area. The second reverses that formula to find height. The third finds one missing parallel side. The fourth uses all four side lengths to reconstruct the height and area when the geometry is determined.

1. 1

   Two bases and height

   Average the parallel side lengths and multiply by the perpendicular height.

2. 2

   Area and two bases

   Divide twice the area by the sum of the parallel sides.

3. 3

   Area, one base and height

   Recover the total base sum, then subtract the known base.

4. 4

   All four sides

   Resolve a leg’s horizontal projection before calculating the height.

Height means the perpendicular distance between the parallel lines. It is not the length of a slanted leg. A genuinely slanted leg is longer than its perpendicular component, so substituting that leg for the height over-estimates the area.

The missing-base mode rearranges the area formula rather than assuming that the two bases are related in some other way. If area is 45, height is 5 and one base is 6, the total base length is 2 × 45 ÷ 5 = 18, so the other base is 12.

Four sides usually determine the height because the difference between the parallel sides constrains the legs’ horizontal projections. Equal parallel sides are different: the figure is a parallelogram, and the same four side lengths can lean through different angles while keeping every side unchanged.

In that equal-base case, no unique height or area exists, but the perimeter still does because all four sides were entered. The calculator marks this result as underdetermined rather than impossible. An impossible result instead means that the supplied lengths cannot form the requested figure.

Terminology depends on region. This page uses the American word trapezoid for a quadrilateral with a pair of parallel sides. In British usage, that figure is called a trapezium. A British trapezoid means the opposite case: a quadrilateral with no parallel sides.

Why four sides can still be insufficient

When the parallel sides have unequal lengths, their difference provides a horizontal offset that constrains the legs. The all-four-sides calculation uses that offset to obtain a right triangle and therefore a height.

When the parallel sides are equal, the offset is zero and the construction becomes a parallelogram. Fixed side lengths do not fix its angle: leaning the figure changes the height and area without changing any side. Perimeter remains the sum of the four supplied lengths.

Trapezoid and trapezium terminology

American and British terminology reverses these names. The American trapezoid handled here has parallel opposite sides and is commonly called a trapezium in Britain. In British usage, a trapezoid instead has no parallel sides and does not fit this calculator’s formulas.

The formula, worked line by line

Let B and b be the parallel side lengths, h their perpendicular separation, A the area, and c and d the two legs. The basic area formula is the average base length multiplied by height.

For the four-side mode, take B as the longer parallel side and set Δ = B − b. A leg’s horizontal projection can then be found from the two leg lengths and Δ.

A = (B + b)h/2
h = 2A/(B + b)
B = 2A/h − b
b = 2A/h − B
P = B + b + c + d
Δ = B − b
x = (Δ² + c² − d²)/(2Δ)
h = √(c² − x²)

The projection formula requires a non-zero base difference. When Δ = 0, fixed side lengths describe a parallelogram whose height varies with its angle, so area is underdetermined.

Perimeter is independent of height but requires every boundary side. Supplying only the parallel sides and height leaves the two leg lengths unknown, so no perimeter is reported.

Questions people ask