Trapezoid Area from Side Lengths Calculator
Find the area, height and perimeter of a trapezoid from its two parallel sides and its two legs — no height needed.
How it works
Height from legs: h = 2·[√(s(s−c)(s−d)(s−|b−a|))] ÷ |b − a|, s = (c + d + |b − a|) ÷ 2 · Area = ½ · (a + b) · h
When you know all four sides of a trapezoid but not its height, slide one leg across so the two legs meet a base of length |b − a|. That forms an ordinary triangle whose area, found by Heron’s formula, equals half the base difference times the trapezoid’s height. Solving back gives the height, and the usual ½(a + b)h then gives the area. If the legs are too short to span the base difference, no such trapezoid exists.
Worked example
For parallel sides 8 and 14 with both legs 5: the base difference is 6, and the triangle with sides 5, 5, 6 has semi-perimeter 8, so its area is √(8·3·3·2) = √144 = 12. The height is 2 × 12 ÷ 6 = 4, and the trapezoid area is ½ × (8 + 14) × 4 = 11 × 4 = 44 square units.
Sources: Trapezoid — Wikipedia
Frequently asked questions
Why do I need the parallel sides to be different?
If both parallel sides are equal the figure is a parallelogram, and its height is not determined by the side lengths alone — it can lean to any angle. The method here relies on a non-zero base difference |b − a| to fix the geometry.
What if the calculator says the sides are invalid?
It means the two legs cannot reach across the base difference to close the shape — the same reason three sticks fail to form a triangle when one is longer than the other two combined. Adjust the leg lengths or the parallel sides.
What is the midsegment?
The midsegment (or median) connects the midpoints of the two legs. Its length is the average of the parallel sides, (a + b) ÷ 2, and multiplying it by the height gives the area directly.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.