What This Calculator Does
You need the area of a trapezoidal plot of land, or the perimeter of a trapezoid cross-section in a machining drawing. This trapezoid calculator takes the two parallel sides and the height, then returns the area, the median (midsegment), and the perimeter. If you leave the non-parallel sides blank, the tool assumes an isosceles trapezoid and computes the legs from the geometry. If you enter the legs, it uses your values for the perimeter.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases (a and b), and the perpendicular distance between them is the height (h). The non-parallel sides are called legs. For a quick refresher on quadrilateral area formulas across shapes, see our Area Calculator.
Inputs Required
- Parallel side a: The length of one base (top)
- Parallel side b: The length of the other base (bottom)
- Height (h): The perpendicular distance between the two bases
- Leg c and Leg d (optional): The non-parallel sides. Leave blank to assume an isosceles trapezoid.
Outputs Provided
- Area: The space enclosed by the trapezoid
- Perimeter: The total length of all four sides
- Median (midsegment): The segment connecting the midpoints of the legs, equal to the average of the two bases
- Legs: The non-parallel sides, computed or entered
How the Calculation Works
Area: A = ½(a + b)h
Median: m = (a + b) / 2
Perimeter: P = a + b + c + d
Isosceles leg: leg = √(((b − a) / 2)² + h²)
The area formula averages the two bases and multiplies by the height. You can think of it as taking a rectangle with width equal to the median and height h. The median is always the arithmetic mean of the two parallel sides, and any line parallel to the bases at height y has length that varies linearly between a and b. That linear variation is why the average works.
For an isosceles trapezoid, the two legs are equal. If you drop perpendiculars from the shorter base to the longer base, you get a rectangle in the middle and two right triangles on the sides. Each right triangle has a horizontal leg of (b − a) / 2 and a vertical leg of h, so the trapezoid leg is the hypotenuse: √(((b − a) / 2)² + h²). For general triangle work, see our Triangle Calculator.
How to Use the Calculator
- Enter the length of parallel side a (the shorter or top base).
- Enter the length of parallel side b (the longer or bottom base).
- Enter the perpendicular height between the two bases. This is not the length of a leg.
- If you know the leg lengths, enter them. Otherwise leave them blank and the calculator assumes an isosceles trapezoid.
- Read the area, perimeter, and median from the results panel.
Example Calculations
Example 1: A Trapezoidal Garden Bed
A landscaper designs a garden bed that runs along a curved driveway. The front edge is 6 feet, the back edge is 10 feet, and the perpendicular depth is 4 feet. The area is ½ × (6 + 10) × 4 = 32 square feet. At a mulch coverage of 3 inches, that is 8 cubic feet of mulch, or about 0.3 cubic yards. The median is (6 + 10) / 2 = 8 feet, which is the width to use if you want to estimate the bed as a rectangle.
Example 2: An Isosceles Trapezoid Window
An architect specifies a trapezoid window with a top of 24 inches, a bottom of 36 inches, and a height of 18 inches. The area is ½ × (24 + 36) × 18 = 540 square inches. The isosceles legs are √(((36 − 24) / 2)² + 18²) = √(36 + 324) = √360 ≈ 18.97 inches. The perimeter is 24 + 36 + 18.97 + 18.97 ≈ 97.95 inches, which tells the glazier how much frame extrusion to cut.
Real World Scenarios
Estimating Paint for a Trapezoid Gable End
A painter bids a job on a house with a trapezoid gable. The bottom (eave) is 28 feet, the top (under the ridge) is 14 feet, and the height is 9 feet. The area is ½ × (28 + 14) × 9 = 189 square feet. One gallon of primer covers about 350 square feet, so a half gallon handles one coat. The painter adds 10% for waste and orders a full gallon to be safe. Misreading the height as a leg length would understate the area and leave the job short on paint.
Sizing a Trapezoidal Ditch Cross-Section
A civil engineer designs a drainage ditch with a trapezoid cross-section. The bottom width is 2 feet, the top width is 6 feet, and the depth is 3 feet. The cross-sectional area is ½ × (2 + 6) × 3 = 12 square feet. At a flow velocity of 2 feet per second, the ditch carries 24 cubic feet per second, or about 10800 gallons per minute. The isosceles side slopes are √(((6 − 2) / 2)² + 3²) = √(4 + 9) = √13 ≈ 3.61 feet, which sets the liner width per side.
Cutting a Trapezoid From a Sheet of Plywood
A carpenter cuts a trapezoid tabletop with a 30-inch front, a 48-inch back, and a 20-inch depth. The area is ½ × (30 + 48) × 20 = 780 square inches. The isosceles legs are √(((48 − 30) / 2)² + 20²) = √(81 + 400) = √481 ≈ 21.93 inches. The perimeter is 30 + 48 + 21.93 + 21.93 ≈ 121.87 inches, which is the length of edge banding needed. Knowing the leg length lets the carpenter set the miter saw angle correctly for the side cuts.
Common Mistakes to Avoid
- Using a leg length instead of the perpendicular height: The area formula needs the perpendicular distance between the bases, not the slanted side. If you only have a leg length, compute the height with h = √(leg² − ((b − a) / 2)²) for an isosceles trapezoid.
- Forgetting that the median equals the average of the bases: The midsegment is (a + b) / 2, not the height. It runs parallel to the bases and is useful for simplifying area to median times height.
- Assuming all trapezoids are isosceles: Only isosceles trapezoids have equal legs. If your trapezoid is scalene, you must enter both leg lengths to get an accurate perimeter. Leaving them blank gives the isosceles approximation.
- Mixing up which sides are parallel: A trapezoid has exactly one pair of parallel sides. Enter those as a and b. The other two sides are the legs.
Limitations of This Calculator
This tool computes area from the two parallel sides and the perpendicular height, which works for any trapezoid. The perimeter is accurate only if you enter the actual leg lengths, or if the trapezoid is isosceles and you leave the legs blank. It does not compute angles, diagonals, or the area of a general quadrilateral. For a right trapezoid (one leg perpendicular to the bases), enter that perpendicular leg as both the height and one of the legs for an exact perimeter.
Authoritative Research and Resources
- Wolfram MathWorld: Trapezoid - A formal reference covering trapezoid properties, the area formula, and the relationship between the median and the bases.
- Maths Is Fun: Trapezoid - A visual explanation of trapezoid area with diagrams showing why the average-of-bases formula works.
For related tools, calculate triangle area with our Triangle Calculator, rectangle and other shape areas with our Area Calculator, or ellipse area with our Ellipse Calculator.