How to Find the Area of a Trapezoid Step by Step

A trapezoid has one pair of parallel sides, often called the bases. If their lengths are a and b and the perpendicular height is h, the area formula is A = one half times (a + b) times h. Another way to say it is: find the average length of the two parallel sides, then multiply by the height.

Suppose the parallel sides are 6 centimetres and 10 centimetres, and the perpendicular height is 4 centimetres. Add the bases to get 16. Multiply by 4 to get 64, then divide by 2. The area is 32 square centimetres. That is the basic process for finding the area of a trapezoid.

The formula works because a trapezoid can be rearranged or paired with another copy to form a parallelogram. The combined base length relates to the sum of the two parallel sides, and dividing by two returns the area of one trapezoid. Seeing that structure makes the formula easier to remember than treating it as a random string of symbols.

The most common mistake is choosing the wrong height. The height is the shortest distance between the parallel sides, measured at a right angle. A sloping edge may be longer than the height and should not be substituted unless the question specifically gives enough information to calculate the perpendicular height from it.

When finding the area of a trapezoid, mark the parallel sides first. They are the pair that go into the addition inside the formula. Then draw or highlight the perpendicular height between them. This small labelling step prevents many errors, especially when the trapezoid is rotated or drawn in an unfamiliar orientation.

Units are another check. If all lengths are in centimetres, area must be in square centimetres. If one length is in metres and another is in centimetres, convert them to a common unit before calculating. Mixing units can produce a number that looks tidy but has no consistent meaning.

Because the trapezoid's area uses the average of the two base lengths, your answer should sit between the areas of two rectangles with the same height and widths equal to the shorter and longer bases. In the example above, those rectangles would have areas of 24 and 40 square centimetres. The trapezoid's 32 square centimetres fits neatly between them.

This estimate is useful when finding the area of a trapezoid because it catches missing division by two. If you had written 64 square centimetres, the result would be larger than even the rectangle built with the longer base, so something would clearly be wrong.

If you learn maths in Britain, you will usually hear this shape called a trapezium, while in the United States it is called a trapezoid. To make things more confusing, some American usage applies the word trapezium to a four-sided shape with no parallel sides at all. The formula does not change with the name. Whatever your textbook calls it, look for exactly one pair of parallel sides, and treat those two sides as the lengths you add together before multiplying by the height.

Sometimes a question hides the height. Imagine an isosceles trapezoid with parallel sides of 10 centimetres and 16 centimetres, and two equal sloping sides of 5 centimetres. The longer base overhangs the shorter one by 6 centimetres in total, or 3 centimetres on each side. Drop a perpendicular line from a top corner, and you create a right-angled triangle with a hypotenuse of 5 and a base of 3. Pythagoras' theorem gives a height of 4. The area is then half of 26 times 4, which is 52 square centimetres.

The trapezoid formula has a surprising second life in calculus and science. Suppose you want the area under a curved graph but cannot calculate it exactly. You can split the region into narrow vertical strips and treat each strip as a trapezoid, with its two parallel sides being the heights of the curve at either edge. Add the areas of all the strips and you have an estimate. In Britain this is called the trapezium rule, and engineers use the same idea with measured data, such as estimating distance travelled from a speed graph.

Try it with the curve y = x squared between x = 0 and x = 2, using two strips of width 1. The heights are 0, 1 and 4. The first strip has area one half times (0 + 1) times 1, which is 0.5, and the second has area one half times (1 + 4) times 1, which is 2.5. The estimate is 3. The exact answer, found by integration, is eight thirds, or about 2.67. The estimate is too large because this curve sags below the straight line joining the tops of each strip, and narrower strips shrink the error.

The trapezoid formula quietly contains other area formulas you already know. If the two parallel sides are equal, say both 7 centimetres, the shape is a parallelogram or a rectangle, and one half times (7 + 7) times h simplifies to 7 times h, which is base times height. If you shrink the shorter side to zero, the top edge disappears and the shape becomes a triangle. With a = 0 and b = 10, the formula gives one half times 10 times h, exactly the triangle formula. Testing these extreme cases is a good way to check that you remember the formula correctly.

There is also a line inside every trapezoid that measures the average base for you. Join the midpoints of the two non-parallel sides and you get the midsegment, sometimes called the median. It runs parallel to the bases, and its length is always half their sum. In the earlier example with bases of 6 and 10 centimetres, the midsegment is 8 centimetres long, and 8 times the height of 4 gives the same 32 square centimetres. If a question gives you the midsegment directly, you can skip the addition and simply multiply it by the perpendicular height.

A diagonal offers another way to see why the formula works. Draw one and the trapezoid splits into two triangles. Both have the same perpendicular height h, since both sit between the same parallel lines. One has base a and the other base b, so their areas are one half a h and one half b h, which add to one half (a + b) h. The formula also runs backwards. If a trapezoid has an area of 45 square centimetres, a height of 6 and one base of 8, then 45 = 3(8 + b), so 8 + b = 15 and the missing base is 7 centimetres.

Engineers meet trapezoids whenever they dig a channel. Canals and drainage ditches are usually cut with sloping banks, because vertical earth sides tend to collapse, so the cross-section of the water is a trapezoid. Suppose a canal is 4 metres wide at the bottom, 8 metres wide at the water surface and 2 metres deep. The water's cross-sectional area is one half times (4 + 8) times 2, or 12 square metres. If the water flows at 0.5 metres per second, multiplying area by speed gives the flow: 6 cubic metres every second. Hydrologists estimate river flow in a similar way, measuring the cross-section in strips.

How to find the area of a trapezoid follows one steady formula: add the two parallel sides, multiply by the perpendicular height, and divide by two. Label the bases, identify the right-angle height, keep units consistent, and check that the final area has square units. A quick size estimate can catch many mistakes before you finish.