A trapezoid is a four-sided shape with one pair of parallel sides, called the bases. The other two sides are the legs, and the perpendicular distance between the bases is the height. This calculator starts from whatever you can measure: the two bases plus the height, both legs, the single leg of an isosceles trapezoid, or the area.
How to use the trapezoid calculator
- Pick an option from the I know the menu: Both bases and the height, Both bases and both legs, Isosceles: both bases and the leg, or Area and both bases (find the height).
- Enter the Bottom base (a) and Top base (b), the two parallel sides. Either one may be the longer.
- Fill in the field that appears: Height (h), Left leg (c) and Right leg (d), Leg (c = d), or Area (A).
- Choose your units. Every mode returns the area, height and midsegment; leg modes add the perimeter, all four angles and both diagonals.
Trapezoid formulas
The term (a + b) ÷ 2 is the midsegment m, the line joining the midpoints of the two legs, so the area is simply midsegment × height. To see why, place a second copy of the trapezoid upside down beside the first. Together they form a parallelogram with base a + b and height h, and the trapezoid is exactly half of it.
How the legs fix the height
Slide the top base sideways until the two legs meet and they form a triangle with sides c, d and the difference of the bases, a − b. That triangle has the same height as the trapezoid. The calculator finds the horizontal offset x of the top-left corner, then applies the Pythagorean theorem:
Take a lot whose parallel front and back lines are 36 ft and 15 ft, with side lines of 13 ft and 20 ft. The offset is ((169 − 400) ÷ 21 + 21) ÷ 2 = 5 ft, so the height is √(169 − 25) = 12 ft and the area is 25.5 × 12 = 306 ft².
In an isosceles trapezoid the offset is simply (a − b) ÷ 2. The leg modes reject equal bases because with a = b the shape is a parallelogram that can lean at any angle, so its legs alone don’t fix the height.
Worked example: a bay-window flower bed
A flower bed in front of a bay window is an isosceles trapezoid: 16 ft along the house, 10 ft along the front edge and 5 ft along each angled side. Choose Isosceles: both bases and the leg and enter a = 16, b = 10, leg = 5, in feet.
- Each leg overhangs the front edge by (16 − 10) ÷ 2 = 3 ft.
- Height: √(5² − 3²) = 4 ft from the house to the front edge.
- Area: (16 + 10) ÷ 2 × 4 = 52 ft². Mulched 3 inches deep, that is 52 × 0.25 = 13 ft³, just under half a cubic yard.
- Perimeter: 16 + 10 + 5 + 5 = 36 ft, or 20 ft of edging if the house wall needs none.
- Angles: 53.13° at the two house corners and 126.87° at the front corners.
- Diagonals: 13.60 ft each, a quick symmetry check when you stake the bed out.
Trapezoid or trapezium, and other definitions
In the US and Canada a trapezoid has a pair of parallel sides and a trapezium has none; in the UK and much of the Commonwealth the meanings are reversed. Textbooks also disagree on whether a trapezoid has exactly one pair of parallel sides or at least one. Under the inclusive definition every parallelogram, rectangle and square is a special trapezoid, and the area formula still works: set a = b and it becomes base × height.
Special cases worth recognizing:
- Isosceles trapezoid: equal legs, equal base angles and equal diagonals, with all four corners on one circle.
- Right trapezoid: one leg is perpendicular to the bases, so that leg equals the height.
- Trapezoidal channels: ditches, canals and embankments usually have a trapezoidal cross-section; multiply its area by the length to estimate the volume of earth.
The trapezoidal rule in calculus uses the same idea, estimating the area under a curve by adding up thin trapezoids.
For shapes with two pairs of parallel sides use the parallelogram calculator, and for a leg or diagonal on its own the Pythagorean theorem calculator is quicker.
Frequently asked questions
What is the difference between a trapezoid and a trapezium?
In American English a trapezoid is a quadrilateral with a pair of parallel sides, and a trapezium has no parallel sides at all. British English swaps the two words, so a UK 'trapezium' is a US 'trapezoid'. The formulas on this page apply to the shape with parallel sides, whichever name your textbook uses.
How do I find the height of a trapezoid?
If you know the area, divide twice the area by the sum of the bases: h = 2A ÷ (a + b). If you know the legs instead, use the Pythagorean theorem. In an isosceles trapezoid with bases 16 and 10 and legs of 5, each leg overhangs by 3, so h = √(5² − 3²) = 4.
Why do I need the legs to get the perimeter?
The bases and height fix the area, but the top base can slide left or right without changing it. Every position gives different legs, angles and perimeter. Entering both legs, or one leg for an isosceles trapezoid, pins the shape down.
Can the top base be longer than the bottom base?
Yes. The labels only say which parallel side is drawn at the bottom. If the top base is longer, the bottom angles come out obtuse instead of acute, while the area and perimeter stay the same.
Why does the calculator say the legs can't close the trapezoid?
Slide the two legs together and they form a triangle whose third side is the difference of the bases. If that triangle cannot exist, neither can the trapezoid. Bases of 12 and 6 with legs of 2 and 9 fail, because 2 + 6 is less than 9.