Trapezoid Calculator

Background.

The area of a trapezoid is half the sum of its parallel sides times the height, which is easy until you notice you never measured the height. Run a tape around a plot of land, a sloping panel or the cross-section of a ditch and what you have is four side lengths. Turning those into a height takes a step most formula sheets leave out, and it is the default here: give the two parallel sides and both slanted legs, and the height, area, perimeter, midsegment, diagonals and base angles all follow from your measurements alone.

The step is needed because two parallel sides and a height do not fix the shape. Slide the shorter side left or right along its line and the area stays the same while the legs, the diagonals and the angles all change. If the height really is what you measured, two other modes fill the gap with an assumption: isosceles, with equal legs, or a right trapezoid, with one leg standing perpendicular to both bases. The area they report is exact either way; the legs, diagonals and angles describe the assumed shape.

Trapezoid is used in the North American sense, a quadrilateral with exactly one pair of parallel sides, the shape British English calls a trapezium. Under that exclusive definition a parallelogram does not qualify, so the longer base must be strictly longer than the shorter one, and four lengths that cannot close into a trapezoid are refused with the reason.

What is trapezoid calculator?

A trapezoid is a four-sided figure with one pair of parallel sides, the bases a (longer) and b (shorter), joined by two legs, c on the left and d on the right. The height h is the perpendicular distance between the bases. The midsegment, the line joining the midpoints of the two legs, runs parallel to the bases and is exactly their average, ½(a + b), however the legs slope. Since the area is that midsegment times the height, a trapezoid covers the same area as a rectangle of the same height whose width is the average of its two bases. An isosceles trapezoid has equal legs, which forces equal base angles and equal diagonals; a right trapezoid has one leg perpendicular to both bases.

How to use this calculator.

1. 01Under "Which measurements do you have?", pick the first option if you know both slanted sides; nothing is then assumed. Choose isosceles or right trapezoid only when the perpendicular height is what you measured.
2. 02Enter base a as the longer parallel side and base b as the shorter. Any unit works if every length uses the same one, and the area comes back in that unit squared.
3. 03With both legs, enter c on the left and d on the right. Swapping them mirrors the shape: the area, perimeter and midsegment stay the same and the two base angles change places.
4. 04With the height, measure it at right angles to the bases rather than along a leg; a slanted leg is always longer than the height.
5. 05To check something already built, compare its diagonals with the results. Equal diagonals mean an isosceles trapezoid; unequal ones mean the legs differ, however symmetric the piece looks.

The formula.

Lay the long base along the x-axis from (0, 0) to (a, 0), and the short base at height h from (p, h) to (p + b, h). Here p is how far the short base is shifted from the left end, and k = a − b is the total overhang. The two legs give c² = p² + h² and d² = (k − p)² + h². Subtracting one from the other removes h and leaves p = (k² + c² − d²) ÷ 2k, and then h = √(c² − p²). A negative p is allowed; it means the top base reaches out past the bottom-left corner, and the left base angle is then greater than 90°.

Everything else is read off the coordinates. Area A = ½(a + b) × h, midsegment m = ½(a + b), perimeter a + b + c + d, diagonals √((p + b)² + h²) and √((a − p)² + h²), and base angles the arctangents of h ÷ p and h ÷ (k − p), in degrees. The isosceles mode sets p = k ÷ 2 and the right-trapezoid mode sets p = 0, and the same formulas then give the legs, diagonals and angles.

Sliding the right leg along until it starts where the left leg starts turns c, d and k into the sides of a triangle, so a trapezoid with these sides exists only if |c − d| < k < c + d, the triangle inequality. Outside that range the height would be the square root of a negative number. Wolfram MathWorld gives a closed form that skips the coordinates, h = η ÷ 2(a − b) with η = √((−a + b + c + d)(a − b + c + d)(a − b + c − d)(a − b − c + d)), and it agrees.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Trapezoid’ (live revision retrieved 2026-07-29). Source for Equation (1) area A = ½(a + b)h, Equation (10) median m = ½(a + b), Equations (17) and (18) for the two diagonals in terms of the four side lengths, and Equation (19) height h = η/(2|b − a|) with η = √((−a+b+c+d)(a−b+c+d)(a−b+c−d)(a−b−c+d)). Also the statement ‘The trapezoid is equivalent to the British definition of trapezium.’ Equations (17), (18) and (19) agree with this calculator's independently derived values. Open access.
2. [2]Wolfram Research, MathWorld — ‘Heron's Formula’ (live revision retrieved 2026-07-29). Equation (2), Δ = √(s(s − a)(s − b)(s − c)) with s the semiperimeter. Used as a third, fully independent derivation of the trapezoid's height: the triangle formed by the two legs and the overhang a − b has area √216 and therefore height 4.898979485566356 on a base of 6, matching the coordinate solution exactly. Open access.
3. [3]Euclid, Elements, Book I, Definition 22 — after defining the square, oblong, rhombus and rhomboid, ‘let quadrilaterals other than these be called trapezia.’ David E. Joyce web edition of the Heath translation, Clark University; quotation confirmed on retrieval 2026-07-29. Cited for the naming history and the definitional scope note, not for any computed value.
4. [4]Wolfram Research, MathWorld — ‘Triangle Inequality’ (live revision retrieved 2026-07-29). The condition that a triangle with sides x, y and z exists only when each side is shorter than the sum of the other two. Applied here to the legs c and d and the overhang a − b, it is the exact existence condition for a trapezoid and is what this calculator enforces before computing anything. Open access.