How to calculate trapezoid area, step by step
To calculate trapezoid area, start with the two parallel sides, which are the bases, and the perpendicular distance between them, which is the height. Add the two bases together, divide the sum by two to get their average, and multiply that average by the height. The average of the bases has a name of its own — the midsegment — and it is the length of the segment joining the midpoints of the two legs. The formula is A = (a + b) / 2 × h, where a and b are the bases and h is the height. Work it out in the trapezoid-area-calculator.
The worked example: bases 8 and 12, height 5
The worked example, step by step. Step 1 — add the bases: 8 + 12 is 20. Step 2 — average them: 20 divided by 2 is 10, and that is the midsegment. Step 3 — multiply by the height: 10 × 5 is 50. Step 4 — write the units: the bases and the height are lengths, so the answer is 50 square units. Step 5 — check it backwards: 50 divided by the height of 5 is 10, which is the midsegment, and twice the midsegment minus the second base is 10 × 2 − 12, which is 8 — the first base, so the working checks out.
Which sides are the bases
A trapezoid is a four-sided shape with one pair of parallel sides, and those two are the bases. The other two sides, the legs, are not in the formula at all. It does not matter which base is longer or which one is on top: the formula adds them, and addition does not care about the order. What matters is that both numbers you add are parallel sides. If you use a leg as a base, the answer is the area of a different shape, and no amount of careful arithmetic will save it. When a diagram labels the sides, look for the parallel marks — the little matching arrowheads — and those are the two numbers that go into the sum.
What the height really is
The height is the perpendicular distance between the two bases — the straight line across, at a right angle to them. It is not the slanted leg. A leg is always at least as long as the height, and usually longer, so if you are given a leg and not a height, you do not have the number the formula needs. Sometimes the height has to be found first, from a right triangle on the side of the shape, or from the area and the two bases using the rearranged formula below. If a diagram draws the height as a dashed line inside the shape, that dashed line is the number to use, and the solid slanted side beside it is not.
How to solve for a missing height
The same formula runs backwards. If you know the area and the two bases, the height is twice the area divided by the sum of the bases: h = 2A / (a + b). With an area of 50 square units and bases 8 and 12, twice the area is 100, the sum of the bases is 20, and 100 divided by 20 is 5 — the height from the worked example. The check is the forward formula: with bases 8 and 12 and a height of 5, the area is 50 square units, so the height is right. If the two bases add to zero, there is no height to find, because a shape with no bases has no area — the calculator refuses that case rather than dividing by zero.
How to solve for a missing base
If you know the area, one base and the height, the other base is twice the area divided by the height, minus the base you know: b = 2A / h − a. With an area of 50 square units, a known base of 8 and a height of 5, twice the area over the height is 20, and 20 minus 8 is 12 — the second base from the worked example. The check is the same forward formula: bases 8 and 12 with a height of 5 give an area of 50 square units. If the solved base comes out below zero, no such trapezoid exists — the area is too small for the base and height you gave it — and the calculator says so instead of a negative length.
When the formula becomes a parallelogram or a triangle
The trapezoid formula has two familiar shapes hiding at its ends. When the two bases are equal, the trapezoid is a parallelogram: the average of two equal numbers is the number itself, so the area is just base times height, exactly as the parallelogram formula says. When one base is zero, the trapezoid is a triangle: the average of a base and zero is half the base, so the area is half of base times height, exactly as the triangle formula says. The trapezoid formula is the general case, and the parallelogram and triangle formulas are what it turns into at the two ends. That is also a useful check: if your trapezoid is nearly a parallelogram, the area should be nearly base times height, and if nearly a triangle, nearly half of base times height.
Common mistakes
The first is using a leg as a base: only the parallel sides go into the sum. The second is using a slanted leg as the height: the height is perpendicular, and a leg overstates the area. The third is forgetting to divide by two: adding the bases and multiplying by the height gives twice the area, not the area. The fourth is adding the height into the average: the average is of the two bases only, and the height multiplies the average afterwards. The fifth is dropping the units: bases and height are lengths, so the area is in square units, and a number without its units is easy to confuse with a length. Work it out in the trapezoid-area-calculator, and check each step against the worked example above.
A quick check you can do in your head
You can sanity-check any trapezoid area without a calculator. The midsegment is the average of the two bases: with bases 8 and 12 it is 10, so any midsegment that is not between the two bases is wrong — the average of two numbers always sits between them. The area is the midsegment times the height: 10 × 5 is 50, so any area that does not split into a midsegment and a height that multiply to it is wrong. And the units always come out squared: an answer in plain units, not square units, is a length, not an area. If a calculator says the area is far above 50 square units on these numbers, or the midsegment is far from 10, the number is wrong — run your numbers through the trapezoid-area-calculator and compare them with the worked example above.