Area of a Trapezoid Calculator

A = (a + b) ÷ 2 × h

a

One of the two parallel sides, in centimetres. Which one you call a and which you call b does not matter — the two are added together, so putting them in the other way round gives the same answer

b

The other parallel side, in centimetres. It may be longer or shorter than a; a top base longer than the bottom base is just the shape upside down

h

The perpendicular distance between the two bases, in centimetres — measured at a right angle to both of them, never along a sloping side

(a + b) ÷ 2

The average of the two bases, and the whole trick of the formula. A trapezoid's area behaves as if the shape were a rectangle whose width is the average of its two bases

Two

The divisor, and the reason this page contains the other two shapes. When the bases are equal the halving and the averaging cancel and base times height comes back; when one base is zero the average is half the other and the triangle's halving appears

Area

How much surface the trapezoid encloses, in square centimetres — always this unit, whatever the dropdowns say

Four decimal places

How wide the reading is written. Halving the sum of two bases is exact for whole centimetres, so a fraction only appears when the inputs themselves carry one

Anything shaped like a trapezoid and priced or covered by area. Roofs are the classic case and the reason the shape has a name in every tradesman's vocabulary: a hipped roof, a gable end seen from the side, the trapezoid between two dormers, all of them are this formula, and roofing is ordered by the square metre. Land is the other old one. A plot with two parallel sides and two that are not is a trapezoid, and a plot with more sides than that is usually cut into trapezoids and triangles, which is how surveyors have measured irregular ground since long before there were calculators. Beyond those two it is the same shape turning up wherever a thing is wider at one end than the other: a tapering driveway, a piece of land between two diverging roads, an acoustic panel, a section of duct that widens, a trapezoidal table top, the face of a dam, a drainage channel. It is also worth knowing for a reason that has nothing to do with trapezoids. Because the equal-bases case gives the parallelogram and the zero-base case gives the triangle, this one formula covers the area of every quadrilateral with a pair of parallel sides — so if you remember only this, you have the other two for free, and the rectangle as well. That is the sense in which this page is the general one.

1. A top base of 6, a bottom base of 10 and a height of 4

   1. Add the two bases: 6 + 10 = 16
   2. Halve to get the average width: 16 ÷ 2 = 8
   3. Multiply by the height: 8 × 4 = 32

   The pair the page loads with. It is also the row to demonstrate that the order of the bases does not matter: put 10 in the top box and 6 in the bottom one, and it comes to 32 again. The two are added before anything else happens, so the boxes are not really a top and a bottom at all — they are just the two parallel sides.

2. A top base of 12, a bottom base of 4 and a height of 7

   1. Add the two bases: 12 + 4 = 16
   2. Halve: 16 ÷ 2 = 8
   3. Multiply by the height: 8 × 7 = 56

   A trapezoid whose top base is longer than its bottom base, which is not an error and is not a different shape in any way that matters — it just means the shape is standing on its longer side. Because the two bases are added before anything else happens, entering 4 on top and 12 underneath reaches the same 56. Try it both ways and watch the answer refuse to change.

3. A top base of 8 and a bottom base of 8

   1. Add the two bases: 8 + 8 = 16
   2. Halve: 16 ÷ 2 = 8
   3. Multiply by the height: 8 × 5 = 40

   The parallelogram line. When the two bases are equal there is nothing left to average, and the formula does what the parallelogram page does: base times height, 40. The halving and the averaging cancel each other out exactly, which is why this page contains that one rather than competing with it.

4. A top base of 0 and a bottom base of 10

   1. Add the two bases: 0 + 10 = 10
   2. Halve: 10 ÷ 2 = 5
   3. Multiply by the height: 5 × 4 = 20

   The triangle line, and the other end of the formula. A top base of zero means the shape has been squeezed to a point, so it is a triangle with a base of 10 and a height of 4 — and the triangle page gives 20 for exactly those two numbers. The average of 0 and 10 is 5, which is half the base, which is the triangle's own halving arriving from the other direction.

5. A top base of 1, a bottom base of 9 and a height of 10

   1. Add the two bases: 1 + 9 = 10
   2. Halve: 10 ÷ 2 = 5
   3. Multiply by the height: 5 × 10 = 50

   A narrow strip on top of a much wider one, which is what a tapering plot or a widening duct actually looks like. The average of 1 and 9 is 5, and 5 is also the average of 0 and 10 — which is the same fact as the row above, seen from a different pair of numbers. Nothing here needed a decimal, and nothing was rounded.

This page computes the area and nothing else. It does not give the perimeter, the sloping sides, the height or the angles, and it will not work backwards from an area to a missing measurement — an area on its own does not determine a trapezoid, so there is nothing to work backwards to. The height is the perpendicular distance between the two bases and not the length of a sloping side; if the only measurement you have runs along one of the sloping edges, it is the wrong number for this box. Both bases and the height must be in the same units, and the answer is always in square centimetres whatever the dropdowns say, so a shape measured in inches comes back as a square-centimetre figure that has to be divided by 6.4516 to be read in square inches. The shape must have exactly one pair of parallel sides: a general quadrilateral with no parallel sides needs to be split into triangles, and a shape with two pairs of parallel sides is a parallelogram, which this will still compute but which has a page of its own. A trapezoid drawn on a curved surface is out of scope, and so is the volume of a solid with trapezoidal faces.

Does it matter which base goes in which box?

No, and it is worth knowing that before you start. The formula adds the two together before it does anything else, so a base of 6 with a base of 10 gives the same answer whichever way round they are typed. A top base longer than the bottom base is not a mistake either — it just means the trapezoid is standing the other way up.

What exactly is the height?

The perpendicular distance between the two parallel sides, measured at a right angle to both. It is not the length of either sloping side, which is always longer whenever the shape leans. This is the same warning as on the triangle page, and for the same reason: entering a sloping side gives a larger area that looks entirely plausible.

Why does the parallelogram page exist if this one covers it?

Because most people who want a parallelogram's area do not think of it as a trapezoid with two equal bases, and making them take that step is work for no benefit. The two pages agree exactly: give this one two bases of 8 and a height of 5 and you get 40, which is what the parallelogram page prints for the same shape. The overlap is deliberate, and each page's table has a row sitting on that point.

What if one of the bases is zero?

Then the trapezoid has been squeezed to a point and it is a triangle, and the answer is right. A top base of zero with a bottom base of 10 and a height of 4 gives 20 — the same 20 the triangle page gives for a base of 10 and a height of 4, because that is the same shape. Zero is treated as a real value rather than an empty box; leaving a box blank is different and shows no result at all.

Can I work out a missing base from the area?

Not from the area alone. An area does not determine a trapezoid — there are infinitely many trapezoids with the same area and very different sides — so there is nothing to solve backwards to. If you know the area, the height and one of the bases, then the other base does follow from the formula by rearranging it, but that is arithmetic you would do by hand rather than something this page offers.

Does this work for a shape with two pairs of parallel sides?

It gives the right number, yes, because a shape with two pairs of parallel sides is a parallelogram — and this formula was already shown to reduce to base times height when the two bases are equal, which is exactly the parallelogram case. But such a shape has a page of its own, and using it is usually clearer than treating a rectangle as a trapezoid.

• Trapezoid — the shape, the vocabulary of bases and legs, and the area formula this page computes — Wolfram MathWorld (United States)
• Parallelogram — what a trapezoid becomes when its two bases are equal, and the base-times-height result it collapses into — Wolfram MathWorld (United States)
• Quadrilateral — the family this shape belongs to, and where the trapezoid sits in it relative to the parallelogram and the rectangle — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); the area of a trapezoid is part of the compulsory-education mathematics curriculum, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部