Area of a trapezoid calculator

The area of a trapezoid is A = ((a + b) / 2) × h, where a and b are the two parallel sides and h is the perpendicular distance between them. Add the parallel sides, halve the total, and multiply by the height. For sides of 6 and 10 and a height of 4: (6 + 10) / 2 = 8, and 8 × 4 = 32 square units.

The average is the whole idea. A trapezoid is wider at one end than the other, and the width changes evenly from 6 to 10, so the shape has the same area as a rectangle of the average width, 8, and the same height. You can see it by cutting off the triangle that sticks out past the average on one side and flipping it into the gap on the other. That average width is a real line in the shape: the midsegment, joining the midpoints of the two slanted sides, which is exactly 8 long here.

The second proof is the doubling one. Take two copies of the trapezoid, turn one upside down and put them side by side. They make a parallelogram whose base is a + b = 16 and whose height is 4, so its area is 64, and one trapezoid is half of that: 32. Forgetting the halving is the first common mistake, and it gives exactly 64.

The other mistake is the height. It is the perpendicular distance between the two parallel sides, not the length of a slanted side, which is always longer unless that side is at a right angle. In an isosceles trapezoid with these measurements each end overhangs by (10 - 6) / 2 = 2, so each slanted side is √(2² + 4²) = √20, about 4.472. Multiplying 8 by 4.472 would overstate the area by about 12%. If you only have the slanted side and the overhang, get the height with the Pythagorean theorem first.

The formula covers its neighbors as special cases. Set a = b and the two sides are equal, the shape is a parallelogram, and the formula becomes b × h. Set a = 0 and the top shrinks to a point, the shape is a triangle, and the formula becomes ½ × b × h: (0 + 10) / 2 × 4 = 20.

It comes up outside the classroom more than most shapes. The cross-section of a ditch, a channel or a road embankment is usually a trapezoid: 10 feet across the top, 6 across the bottom and 4 deep is 32 square feet, and multiplying by the length gives the cubic feet of soil. A garden bed against an angled fence and a lot whose two side lines run parallel are the same calculation. In the UK the shape is called a trapezium, and the formula is identical.