Trapezoid Side Calculator
Enter any three sides of a trapezoid plus the perimeter to instantly find the missing side. Switch to full-shape mode to also get the area, height, diagonals, angles, and the median line. All results include step-by-step working so you can follow every calculation.
Formula
Worked example
Trapezoid with a = 12 cm, b = 8 cm, c = 6 cm, perimeter = 36 cm: missing side d = 36 - 12 - 8 - 6 = 10 cm. For the full shape with h = 4 cm: area = (12+8)/2 * 4 = 40 cm^2, median = 10 cm.
What is a trapezoid?
A trapezoid (also called a trapezium in British English) is a four-sided polygon with exactly one pair of parallel sides, called the bases. The longer parallel side is usually labelled a, the shorter one b. The two non-parallel sides are called the legs (or lateral sides), labelled c and d. The perpendicular distance between the two bases is the height h. A special case - the isosceles trapezoid - has equal legs (c = d), equal base angles, and equal diagonals. A right trapezoid has two right angles on the same side, so one leg is perpendicular to both bases.
How to find a missing side
If you know the total perimeter P and three of the four sides, the fourth side follows directly from the perimeter formula: P = a + b + c + d, rearranged to d = P - a - b - c. For example, a trapezoid with a = 12, b = 8, c = 6, and P = 36 has a missing leg d = 36 - 12 - 8 - 6 = 10. The same rearrangement works whichever side is unknown; just subtract the three known sides from the perimeter.
Full shape properties: area, height, diagonals, and angles
Once all four sides and the height are known, the remaining properties fall out of standard geometry. Area: A = (a + b) / 2 * h, the average of the two bases multiplied by the height. Median: m = (a + b) / 2, the length of the segment connecting the midpoints of the two legs - it equals the average of the bases. The interior angles at the long base come from the sine of the height divided by the respective leg: alpha = arcsin(h / c) and beta = arcsin(h / d). Diagonals are computed by placing the trapezoid in a coordinate system: the long base along the x-axis, the left leg rising to the short base, then calculating the straight-line distance between opposite corners.
Units and practical tips
This calculator works in millimetres, centimetres, metres, inches, feet, and yards - switch the unit selector before entering your measurements. All internal arithmetic is done in metres to avoid rounding errors from unit conversions. Area results are in square units of whatever length unit you chose. When measuring a real trapezoid, verify that your height is truly perpendicular to both parallel sides - a slanted measurement will underestimate h and produce incorrect area and angle results. For very precise carpentry or engineering work, convert all field measurements to a consistent unit before entering them.
Common trapezoid types and their properties
| Type | Defining property | Key shortcut |
|---|---|---|
| General trapezoid | Exactly one pair of parallel sides | No special symmetry |
| Isosceles trapezoid | Non-parallel legs are equal | Base angles equal; diagonals equal |
| Right trapezoid | Two right angles on one side | One leg equals the height |
| Parallelogram | Both pairs of sides are parallel | Opposite sides equal; special case |
Understanding which type of trapezoid you have helps select the right formula.
Frequently asked questions
How do I find the missing side of a trapezoid?
Use the perimeter formula P = a + b + c + d. Rearrange it to isolate the unknown: d = P - a - b - c. Simply enter the perimeter and the three known sides in this calculator and the fourth side is computed instantly.
What is the difference between a trapezoid and a trapezium?
In American English a trapezoid has exactly one pair of parallel sides. In British English the same shape is called a trapezium. The words are used interchangeably in this calculator.
How is the area of a trapezoid calculated?
Area = (a + b) / 2 * h, where a and b are the two parallel bases and h is the perpendicular height between them. Switch to Full Shape mode and enter all four sides plus the height to have this calculated automatically.
What is the median of a trapezoid?
The median (or midsegment) is the line segment that connects the midpoints of the two legs. Its length is always (a + b) / 2 - the arithmetic average of the two bases. It is also parallel to both bases and exactly half their combined length.
How are the diagonals of a trapezoid calculated?
Place the trapezoid in a coordinate system with the long base along the x-axis. Knowing the height and the horizontal run of each leg (from Pythagoras: run = sqrt(leg^2 - height^2)), you can find the coordinates of all four corners and then apply the distance formula to each diagonal.
What makes a trapezoid isosceles?
An isosceles trapezoid has equal legs (c = d). As a result, the base angles at the long base are equal, the base angles at the short base are equal, and the two diagonals are the same length. The calculator flags this automatically when the legs you enter are equal.
Can I find the height if it is not given?
If you know all four sides you can recover the height using Heron-like algebra, but that requires solving a system of equations. For that calculation, try a dedicated full trapezoid solver. If you have the area and both bases, the height is simply h = 2A / (a + b).