Trapezoid Calculator
Area, perimeter and midsegment from the two parallel sides.
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- Midsegment—
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How to use this trapezoid calculator
- Choose what you have measured. The two parallel sides plus the perpendicular height is the straightforward case. If you only have the four side lengths, switch to that mode and the height is worked out for you.
- Enter the measurements in a single unit.
- Read the area, with the midsegment and the perimeter underneath.
Trapezoid or trapezium?
This shape has the most confusing naming in geometry, and it is worth clearing up before anything else.
| Term | In American English | In British English |
|---|---|---|
| Trapezoid | Exactly one pair of parallel sides | No parallel sides at all |
| Trapezium | No parallel sides at all | Exactly one pair of parallel sides |
The two words mean the opposite things on either side of the Atlantic. This page uses the American convention, so trapezoid here means the shape with one pair of parallel sides, which is what British readers call a trapezium. The formula is the same whichever name you use.
The trapezoid area formula
Average the two parallel sides, then multiply by the perpendicular height. That is all there is to it.
The reason it works is worth a moment. Take a copy of the trapezoid, turn it upside down, and fit it against the original. The two together form a parallelogram whose base is a + b and whose height is h. Its area is (a + b) × h, and your trapezoid is exactly half of that.
Put another way: a trapezoid has the same area as a rectangle of the same height whose width is the average of the two parallel sides. If the parallel sides are 8 and 14, the trapezoid covers the same ground as a rectangle 11 wide.
The height must be perpendicular
As with every area formula, h is the straight-up distance between the two parallel sides, not the length of a sloping leg. Measuring the slanted side instead inflates the answer.
A worked example
A trapezoid has parallel sides of 8 cm and 14 cm with a height of 6 cm.
- Average the parallel sides. (8 + 14) ÷ 2 = 11 cm
- Multiply by the height. 11 × 6 = 66 cm²
That averaged figure of 11 cm is also the midsegment, the line joining the midpoints of the two sloping legs. It is always halfway in both position and length, which gives a useful shortcut: measure the midsegment, multiply by the height, and you have the area without needing either parallel side.
Finding the area from four sides
If the height cannot be measured directly, the four side lengths are enough, provided you know which pair is parallel. The method splits the shape into a triangle and a parallelogram: slide one leg across and you get a triangle whose sides are the two legs and the difference between the parallel sides. Heron’s formula gives that triangle’s area, from which the height follows.
Two things can go wrong, and the calculator reports both. If the difference between the parallel sides is longer than the two legs added together, no such shape exists. And if the parallel sides are equal, the figure is a parallelogram that can lean to any degree, so the four sides alone do not determine a height.
Types of trapezoid
- Isosceles trapezoid — the two legs are equal. It is symmetrical, the base angles match, and the diagonals are equal. Bridges and lampshades are usually this shape.
- Right trapezoid — two adjacent angles are 90 degrees, so one leg is itself the height. Common in cross-sections of ramps and retaining walls.
- Scalene trapezoid — all four sides differ, with no symmetry.
Where the trapezoid rule comes up
The formula extends well beyond the shape itself. The trapezoidal rule in calculus estimates the area under a curve by slicing it into thin trapezoids and adding them, which is more accurate than using rectangles because a trapezoid follows a sloping curve far better than a flat-topped block.
- Land surveying. Irregular plots are commonly broken into trapezoidal strips and summed.
- Earthworks. Drainage channels, embankments and road cuttings have trapezoidal cross-sections, so volume is the area times the length.
- Numerical integration. Any software estimating an integral from sampled data is almost certainly adding up trapezoids.
- Signal processing. Energy under a waveform is found by the same method applied to sampled points.
Frequently asked questions
What is the formula for the area of a trapezoid?
Add the two parallel sides, halve the result, then multiply by the perpendicular height: A = ((a + b) / 2) x h. In words, the area equals the average of the parallel sides times the height.
What is the difference between a trapezoid and a trapezium?
The words swap meaning by country. In American English a trapezoid has one pair of parallel sides; in British English that shape is a trapezium. This page uses the American convention, but the area formula is identical either way.
What is the area of a trapezoid with sides 8 and 14 and height 6?
It is 66 square centimetres. Average the parallel sides to get (8 + 14) / 2 = 11, then multiply by the height of 6.
Can I find the area of a trapezoid without the height?
Yes, if you know all four sides and which pair is parallel. The height is recovered from the two legs and the difference between the parallel sides using Heron’s formula. Switch this calculator to the four-side mode.
What is the midsegment of a trapezoid?
The midsegment joins the midpoints of the two sloping legs. Its length equals the average of the two parallel sides, and it sits exactly halfway up. Multiplying the midsegment by the height also gives the area.
What is an isosceles trapezoid?
One whose two non-parallel legs are equal in length. It is symmetrical about a vertical line, its base angles are equal, and its diagonals are the same length.
Why does averaging the parallel sides work?
Place an upside-down copy of the trapezoid against the original and the two form a parallelogram of base a + b and height h. The trapezoid is half of that, which is the same as averaging the parallel sides and multiplying by the height.
What is the trapezoidal rule?
A method for estimating the area under a curve by dividing it into narrow trapezoids and adding their areas. It is more accurate than using rectangles because a sloping top follows a curve more closely than a flat one.