How to Find the Area of a Trapezoid

A trapezoid is a rectangle that changes width as it goes. Start at 12 across the top, finish at 20 across the bottom, and because the change is steady the average width is simply the mean of the two ends. Once you see it that way the formula stops being something to memorise: it is average width times height, which is what every area calculation has been since the rectangle.

The mean of the parallel sides is the whole idea

Draw a line across the middle of a trapezoid, joining the midpoints of the two sloping sides. That line is called the midline, and its length is exactly the mean of the two parallel sides. For 12 and 20 it measures 16, every time, whatever the sloping sides are doing. The trapezoid has the same area as a rectangle 16 wide and 7 tall, because the triangle of material sticking out below the midline is exactly the triangle missing above it.

That is also why the formula does not care whether the shape leans. A right trapezoid with one vertical side and a symmetric one with two equal slopes have the same area as long as a, b and h match. The lean moves material sideways without adding any, the same way it does in a parallelogram.

Why halving the sum works

Take a second copy of the trapezoid, turn it through half a turn and slide it against the first. The two interlock perfectly, because the sloping side of one matches the sloping side of the other. What you get is a parallelogram, and its base is a plus b.

The parallelogram is 32 wide and 7 tall, so it holds 224 square units. It is built from two identical trapezoids, so one of them holds 112. Divide a plus b by two and you have done exactly that, just in a different order.

The formula also quietly contains two shapes you already know. Set a equal to b and the mean is just that side, so it becomes a parallelogram at b times h. Shrink a to zero and the mean is half of b, so it becomes a triangle at b h over 2. One formula, three shapes, which is a fair reason to learn this one properly rather than the other two separately.

A worked check you can repeat on paper

Four routes, one number. The shoelace check is worth running once for yourself, because it uses nothing but the corner coordinates and so it does not borrow the height, the midline or the formula. If your own trapezoid disagrees between these routes, the measurement is wrong rather than the arithmetic.

When all you have is the four sides

This is the case that sends people hunting, and the honest answer is that it is recoverable but awkward. Call the parallel sides a and b, and the two sloping sides c and d. Slide one sloping side across until both meet, and you have a triangle whose sides are b minus a, c and d. Run Heron on that triangle, and its height is the height of the trapezoid.

Written out, with k standing for b minus a and s for half of k plus c plus d, the height is 2 over k, times the square root of s(s − k)(s − c)(s − d). For the trapezoid above, with sloping sides of 8.0623 each, that returns a height of exactly 7. It also works on the right angled version, where the sides are 7 and 10.6301, and returns 7 again.

That division is the thing to watch. As a approaches b the divisor shrinks toward zero, so a small error in a measured side gets multiplied into a large error in the height. At a equals b it breaks completely, which is not a flaw in the algebra: a parallelogram of given side lengths can lean to any angle, so its area genuinely is not determined by its sides. Four side lengths alone do not fix a four sided shape.

Mistakes that catch people out

• Using a sloping side as the height. The most common error by a distance. The slope is always longer than the perpendicular gap, so the area always comes out too big. In the worked example the slope is 8.06 against a true height of 7, which inflates the answer by 15 percent.
• Adding all four sides before halving. Only the two parallel sides go into the sum. The sloping sides never appear in the formula at all, which is exactly why two trapezoids that look nothing alike can have identical areas.
• Averaging the wrong pair. In a trapezoid drawn on its side, the parallel pair is the left and right, not the top and bottom. Identify which pair is parallel before reaching for a and b.
• Forgetting that the height is measured between the parallel sides. Not from a corner, not along an edge, and not the longest distance across the shape. Perpendicular, from one parallel side to the other.
• Mixing units. Sides in centimetres with a height in metres produces a number that means nothing. Convert first, and remember area scales with the square, so a factor of 100 in length becomes 10,000 in area.

Try these

Common questions

What is the formula for the area of a trapezoid?

Add the two parallel sides, halve the total and multiply by the perpendicular height. In symbols, A = (a + b) h / 2. Sides of 12 and 20 with a height of 7 give 112 square units.

How do you find the area of a trapezoid without the height?

Recover the height from the four side lengths first, using Heron on the triangle whose sides are b minus a, c and d. It works, but it divides by b minus a, so it fails when the parallel sides are equal and becomes unreliable when they are close. If you can measure the height directly, do that instead.

Is the height the same as the sloping side?

No, and confusing the two is the most common way this calculation goes wrong. The height is the perpendicular distance between the parallel sides. The sloping side is longer, and using it in place of the height always overstates the area.

What is the midline of a trapezoid?

The segment joining the midpoints of the two sloping sides. Its length is always the mean of the parallel sides, so for sides of 12 and 20 the midline is 16. The area is simply the midline times the height.

What is the difference between a trapezoid and a trapezium?

Naming, not geometry. In North America a trapezoid has one pair of parallel sides. In British usage that shape is a trapezium, and the word trapezoid means a quadrilateral with no parallel sides at all. The formula on this page applies to the shape with one parallel pair, whichever name your textbook uses.

How do you find the height from the area?

Rearrange to h = 2A / (a + b). An area of 90 with parallel sides of 7 and 11 gives a height of 180 over 18, which is 10.

Related lessons and calculators

• Area formulas for every common shape, where this formula sits alongside the rest, with a calculator covering fifteen shapes.
• Area of a triangle, the shape a trapezoid becomes when the shorter parallel side shrinks to nothing.
• Height of a triangle from area and base, the same rearrangement applied to the simpler shape.
• Area of a triangle from coordinates, which explains the shoelace method used as the third check above.
• Area of 2D shapes calculator, for running trapezoids and the rest without doing the arithmetic yourself.
• the National Council of Teachers of Mathematics, for curriculum guidance on how this material is taught and sequenced.
• Area of a parallelogram, what this shape becomes when the two parallel sides are the same length.
• Area of a rhombus, another quadrilateral where the easiest measurement to take is not the one in the usual formula.
• Height of a trapezoid, the same formula rearranged for when the area is the thing you already know.
• Perimeter of a trapezoid, the measurement the area can never give you, however many sides you know.