How to Find the Height of a Trapezoid

The height of a trapezoid is the perpendicular distance between the two parallel sides, and almost every wrong answer starts by measuring something else. The sloping side is longer, the diagonal is longer still, and a drawing rarely makes it obvious which line is which. Get that straight and the rest is one division.

The height is the gap, and nothing else is

Lay a trapezoid flat with its parallel sides horizontal. The height is how far you have to climb to get from the lower one to the upper one, measured straight up. Slide the top side left or right as much as you like and the height does not change, which is exactly why the area does not change either.

Drop a perpendicular from each end of the shorter parallel side and the shape falls apart into pieces you already know: a right triangle, a rectangle, and another right triangle. In the worked example those bases measure 5, 10 and 9, which add to 24, the longer parallel side. That decomposition is not just a picture. It is the method you use when nobody gives you an area.

A worked check you can repeat on paper

Six routes, one height, and the two runs of 5 and 9 add to 14, which is the difference between the parallel sides. That is the arithmetic the picture was showing: the longer parallel side is the shorter one plus whatever the two sloping ends spread out sideways.

If you want the trapezoid from the area lesson instead, it has parallel sides of 12 and 20 and an area of 112, so its height is 224 over 32, which is 7. Same single division, different numbers.

When you only have the four sides

This is the case that sends people looking for a formula, and there is one. Call the difference between the parallel sides b minus a. The horizontal run of the shorter leg is x = ((b − a)² + c² − d²) divided by 2(b − a), and then the height is the square root of c² minus x². For the worked trapezoid that is (196 + 169 − 225) over 28, which is 140 over 28, which is 5, and √(169 − 25) is 12.

Worth knowing why that works at all. Four side lengths do not pin down a general quadrilateral, which is why a hinged four bar linkage can flop about without any side changing length. Adding the fact that one pair is parallel takes all that freedom away, and the shape becomes rigid. So a trapezoid really is determined by its four sides, and a random quadrilateral is not.

Two things the formula will tell you if you let it. If c² minus x² comes out negative, no trapezoid with those four sides exists: try 10 and 24 with two legs of 5 and you are asked for the square root of a negative number, because two legs of 5 cannot span a 14 unit difference. And if x comes out negative, the trapezoid simply leans the other way, with the top overhanging the bottom. Sides of 10, 14, 13 and 15 give x = −5 and a perfectly real height of 12.

The two shapes that make it easier

An isosceles trapezoid has two equal legs, so the overhang is shared evenly and each run is half the difference of the parallel sides. Parallel sides of 8 and 18 with legs of 13 give runs of 5 each, so the height is √(169 − 25), which is 12 again. No general formula needed, just halve and use Pythagoras.

A right trapezoid is easier still, because one leg is already perpendicular. That leg is the height, read straight off with no working at all. The worked trapezoid turned into a right trapezoid would have a vertical leg of 12 and a sloping one of 18.44, and only one of those two numbers is any use.

What you know	Route to the height	Worked value
Area and both parallel sides	2A divided by a plus b	12
Area and the midline	area divided by the midline	12
All four sides	find the run x, then Pythagoras	12
Isosceles, legs and parallel sides	run is half of b minus a	12
A leg and its angle to the base	leg times the sine of the angle	12
Right trapezoid	the perpendicular leg is the height	12

Mistakes that catch people out

• Using a leg as the height. On the worked trapezoid, putting 13 where 12 belongs turns an area of 204 into 221. That is 8 percent out, small enough to survive a glance, which is exactly what makes it worse than an obvious blunder.
• Forgetting to double the area. The area formula halves the sum of the parallel sides, so undoing it means doubling the area first. Dividing 204 by 34 gives 6, which is half the real height.
• Averaging the wrong pair. Only the parallel sides go into a and b. The legs never enter that sum, no matter how tempting it is on a shape where all four sides are given.
• Assuming both perpendiculars land inside the base. On a trapezoid that leans hard, one foot falls outside and its run is negative. The algebra handles it without complaint, but a sketch drawn the other way will not match your working.
• Trusting a diagram’s height line. Diagrams routinely draw a line from a top corner to the base and label it h when it is really a leg. The height meets the base at a right angle, and if there is no right angle mark, do not assume one.

Try these

Common questions

How do you find the height of a trapezoid from the area?

Double the area and divide by the sum of the parallel sides. An area of 204 with parallel sides of 10 and 24 gives 408 over 34, which is a height of 12. If you have the midline instead, just divide the area by it.

Is the height the same as the slanted side?

No. The height is the perpendicular distance between the parallel sides and is always the shortest of those lines. The worked trapezoid has legs of 13 and 15 and a height of 12, three different numbers for three different lines.

How do you find the height of a trapezoid with only the four sides?

Work out the horizontal run of one leg first: x = ((b − a)² + c² − d²) over 2(b − a). Then the height is the square root of c² minus x². For sides of 10, 24, 13 and 15 the run is 5 and the height is 12.

What is the height of an isosceles trapezoid?

Halve the difference between the parallel sides to get the run, then use Pythagoras on a leg. Parallel sides of 8 and 18 with legs of 13 give a run of 5, so the height is 12.

Can four side lengths give more than one trapezoid?

No, once you know which pair is parallel the shape is fixed. That is unusual for a quadrilateral, since four sides on their own leave a shape free to flex. The parallel condition removes that freedom.

What if the height comes out as the square root of a negative number?

Then no such trapezoid exists and the measurements are wrong. Two legs of 5 cannot bridge parallel sides of 10 and 24, because the difference of 14 is more than the two legs can spread between them.

Related lessons and calculators

• Area of a trapezoid, the formula this lesson runs backwards, with the midline explained properly.
• Height of a triangle from area and base, the same rearrangement on the shape a trapezoid becomes when one parallel side shrinks to nothing.
• Missing side of a right triangle, the Pythagoras step on its own, for when the four sides route is the one giving trouble.
• Area of a parallelogram, the shape a trapezoid becomes when both parallel sides match.
• Area of 2D shapes calculator, for trapezoids given as four sides without the arithmetic.
• Area formulas for every common shape, the index this lesson sits inside.
• Wolfram MathWorld on the trapezoid, for the general treatment and the naming differences between countries.
• Perimeter of a trapezoid, which needs both legs named rather than just the gap between the parallel sides.