Trapezoid Calculator
A trapezoid (or trapezium) has exactly one pair of parallel sides. Pick a solve mode, enter the values you know, and get the area, perimeter, median, angles, and diagonals instantly, with a step-by-step breakdown showing exactly how every number was reached.
Formula
Worked example
For a trapezoid with bases a = 10 cm and b = 6 cm, height h = 5 cm, and legs c = d = 5.39 cm: the median is (10 + 6) / 2 = 8 cm, the area is 8 x 5 = 40 cm squared, and the perimeter is 10 + 6 + 5.39 + 5.39 = 26.78 cm. Angle A = arcsin(5 / 5.39) = arcsin(0.927) = about 68 degrees.
How the trapezoid area formula works
A trapezoid has exactly one pair of parallel sides, called the bases, labelled a (longer) and b (shorter). Its area is the average of the two bases multiplied by the perpendicular height between them: A = (1/2)(a + b) h. The intuition is geometric: if you rotate a copy of the trapezoid 180 degrees and join it to the original, the two pieces form a parallelogram whose base is a + b and whose height is h. That parallelogram has area (a + b) h, so your single trapezoid is exactly half of it. Because both bases enter the formula symmetrically, it does not matter which you call a and which you call b. The height must be perpendicular to the bases, measured straight across, not along a slanted leg.
Median, legs, and perimeter
The median, also called the midsegment, connects the midpoints of the two non-parallel legs. It always equals (a + b) / 2, runs parallel to both bases, and lets you rewrite the area as median x height. The two slanted legs, c and d, are the non-parallel sides. In an isosceles trapezoid they are equal; in a general (scalene) trapezoid they differ. The legs affect the perimeter but never the area. The perimeter is simply a + b + c + d. If you know the height and one leg, you can derive the other quantities using trigonometry: the base angle A satisfies sin(A) = h / c, and the horizontal offset of the leg is c cos(A).
Angles and diagonals
A trapezoid has four interior angles. Because the two bases are parallel, the angles on the same leg are co-interior (also called same-side interior or consecutive) angles and always sum to 180 degrees: angle A + angle B = 180 degrees and angle C + angle D = 180 degrees. Given the two base angles A and D, the height follows from the formula h = (a - b) / (cot A + cot D). The two diagonals p and q are found using coordinate geometry: place base a along the x-axis, compute the corner coordinates from the legs and angles, then apply the distance formula. In an isosceles trapezoid the diagonals are equal in length.
Reverse-solve modes
Sometimes you know the area and need a missing dimension. If you know the area and both bases, switch the solve mode to "find height" and the calculator rearranges A = (1/2)(a + b) h to give h = 2A / (a + b). If you know the area, one base, and the height, switch to "find base a" and the calculator gives a = 2A / h - b. These reverse-solve modes are useful in flooring, land-area problems, and classroom exercises where the final measurement is given and you need to back out a side.
Trapezoid quick-reference examples
| Base a | Base b | Height h | Median | Area | Perimeter (c=d=5.39) |
|---|---|---|---|---|---|
| 10 | 6 | 5 | 8 | 40 | 26.78 |
| 12 | 8 | 4 | 10 | 40 | 28 + 2c |
| 20 | 14 | 9 | 17 | 153 | 34 + 2c |
| 7 | 3 | 6 | 5 | 30 | 10 + 2c |
| 15 | 9 | 7 | 12 | 84 | 24 + 2c |
| 8 | 4 | 3.5 | 6 | 21 | 12 + 2c |
Areas use A = (1/2)(a + b) h. Diagonals are for isosceles cases (c = d).
Frequently asked questions
What is the formula for the area of a trapezoid?
The area of a trapezoid is A = (1/2)(a + b) h, where a and b are the two parallel sides (bases) and h is the perpendicular height between them. Equivalently, it is the midsegment (median) times the height. The legs do not affect the area at all.
How do I find the height of a trapezoid if I only know the area and the bases?
Rearrange the area formula to h = 2A / (a + b). For example, if the area is 40 square centimeters and the bases are 10 and 6 cm, then h = 2 x 40 / (10 + 6) = 80 / 16 = 5 cm. Use the "find height" solve mode in this calculator to do it automatically.
Is the height the same as the leg length?
Not usually. The height is the perpendicular distance between the two parallel bases, measured at a right angle to them. The leg is the actual slanted side and is always longer than the height unless the trapezoid has a right angle at that corner. If leg c makes angle A with the base, then h = c sin(A) and the horizontal offset is c cos(A).
How do I find the angles of a trapezoid?
If you know the height and one leg, the base angle is A = arcsin(h / c). Because the two parallel sides make co-interior angles on each leg, the opposite base angle is B = 180 degrees minus A. Switch the calculator to the "all properties from four sides" or "a, b, c + angle A" mode and all four angles are computed for you.
What is the difference between a scalene, isosceles, and right trapezoid?
A scalene (general) trapezoid has legs of different lengths and no right angles. An isosceles trapezoid has two equal legs, equal base angles, and equal diagonals. A right trapezoid has exactly one right angle: one leg is perpendicular to the bases, so its length equals the height, while the other leg is slanted. The area formula is the same for all three types.
How are the diagonals of a trapezoid calculated?
Place the longer base a along the x-axis with the left corner at the origin. The top-left corner is at (c cos(A), h) and the top-right corner is at (c cos(A) + b, h). Diagonal p runs from the bottom-left origin to the top-right corner; diagonal q runs from the bottom-right (a, 0) to the top-left corner. Apply the Pythagorean theorem (distance formula) to each. In an isosceles trapezoid p and q are equal.