Trapezoid Calculator
Area from the parallel sides and height, or the height from all four sides.
Updated
m
m
m
m — for the perimeter
m — for the perimeter
Trapezoid, or trapezium
50m²
The two parallel sides average 10 m, across a perpendicular height of 5 m.
- Parallel side a
- 12 m
- Parallel side b
- 8 m
- Height
- 5 m
- Midsegment
- 10 m
- Area
- 50 m²
- Perimeter
- needs both legs
The perimeter is blank on purpose. Two parallel sides and a height fix the area but say nothing about how long the sloping legs are — infinitely many trapezoids share this area with different perimeters.
The height must be the perpendicular distance between the parallel sides, not the length of a slanted leg. Using the leg is the commonest wrong answer for this shape, and because the leg is always the longer, it always over-estimates.
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In short
What can the trapezoid calculator solve?
It supports 4 modes: area from 2 parallel sides and height, height from area and both parallel sides, a missing parallel side, or height and area from all 4 sides. Perimeter appears only when both legs were supplied.
A slanted leg is not the perpendicular height.
How to use the trapezoid calculator
Match the inputs to one of four modes
The first mode uses both parallel sides and their perpendicular height to find area. The second reverses that formula to find height. The third finds one missing parallel side. The fourth uses all four side lengths to reconstruct the height and area when the geometry is determined.
- 1
Two bases and height
Average the parallel side lengths and multiply by the perpendicular height.
- 2
Area and two bases
Divide twice the area by the sum of the parallel sides.
- 3
Area, one base and height
Recover the total base sum, then subtract the known base.
- 4
All four sides
Resolve a leg’s horizontal projection before calculating the height.
Height means the perpendicular distance between the parallel lines. It is not the length of a slanted leg. A genuinely slanted leg is longer than its perpendicular component, so substituting that leg for the height over-estimates the area.
The missing-base mode rearranges the area formula rather than assuming that the two bases are related in some other way. If area is 45, height is 5 and one base is 6, the total base length is 2 × 45 ÷ 5 = 18, so the other base is 12.
- Parallel sides
- B = 10, b = 4
- Legs
- c = 5, d = 5
- Base difference
- Δ = 10 − 4 = 6
- Horizontal projection
- x = (6² + 5² − 5²) ÷ 12 = 3
- Height
- h = √(5² − 3²) = 4
- Area and perimeter
- A = 28; P = 24
Area is (10 + 4) × 4 ÷ 2 = 28, while all four sides sum to 24.
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Four sides usually determine the height because the difference between the parallel sides constrains the legs’ horizontal projections. Equal parallel sides are different: the figure is a parallelogram, and the same four side lengths can lean through different angles while keeping every side unchanged.
(B + b)h/2
Area
Average base length times height
B + b + c + d
Perimeter
Available only with both legs
Underdetermined
Equal-base four-side case
Perimeter remains available
In that equal-base case, no unique height or area exists, but the perimeter still does because all four sides were entered. The calculator marks this result as underdetermined rather than impossible. An impossible result instead means that the supplied lengths cannot form the requested figure.
Do
- Identify the pair of opposite parallel sides before entering values.
- Use the perpendicular distance as the height.
- Read the perimeter-available flag before displaying a perimeter.
- Distinguish underdetermined geometry from incompatible measurements.
Don't
- Substitute a slanted leg for the perpendicular height.
- Infer perimeter from two bases and height alone.
- Invent an area when equal bases leave the lean angle free.
Terminology depends on region. This page uses the American word trapezoid for a quadrilateral with a pair of parallel sides. In British usage, that figure is called a trapezium. A British trapezoid means the opposite case: a quadrilateral with no parallel sides.
Compare with a rectangle
Use the rectangle calculator when every corner is a right angle and opposite sides match.
Open rectangle calculator →Worked trapezoid cases across the four supported modes and the equal-base exception
| Mode and known values | Checkable working | Result |
|---|---|---|
| Mode 1: B = 14, b = 8, h = 5 | A = (14 + 8) × 5 ÷ 2 = 22 × 5 ÷ 2 | A = 55; perimeter unavailable |
| Mode 1: B = 7, b = 3, h = 4 | A = (7 + 3) × 4 ÷ 2 = 10 × 4 ÷ 2 | A = 20; perimeter unavailable |
| Mode 2: A = 54, B = 11, b = 7 | h = 2 × 54 ÷ (11 + 7) = 108 ÷ 18 | h = 6; perimeter unavailable |
| Mode 2: A = 30, B = 6, b = 4 | h = 2 × 30 ÷ (6 + 4) = 60 ÷ 10 | h = 6; perimeter unavailable |
| Mode 3: A = 45, b = 6, h = 5 | B = 2 × 45 ÷ 5 − 6 = 18 − 6 | B = 12; perimeter unavailable |
| Mode 3: A = 24, B = 10, h = 3 | b = 2 × 24 ÷ 3 − 10 = 16 − 10 | b = 6; perimeter unavailable |
| Mode 4: B = 10, b = 4, c = 5, d = 5 | Δ = 6; x = (6² + 5² − 5²)/(2 × 6) = 3; h = √(5² − 3²) | h = 4; A = 28; P = 24 |
| Mode 4: B = 13, b = 5, c = 5, d = 5 | Δ = 8; x = (8² + 5² − 5²)/(2 × 8) = 4; h = √(5² − 4²) | h = 3; A = 27; P = 28 |
| Mode 4: B = 8, b = 8, c = 5, d = 5 | Δ = 8 − 8 = 0; P = 8 + 8 + 5 + 5 | Underdetermined height and area; P = 26 |
Why four sides can still be insufficient
When the parallel sides have unequal lengths, their difference provides a horizontal offset that constrains the legs. The all-four-sides calculation uses that offset to obtain a right triangle and therefore a height.
When the parallel sides are equal, the offset is zero and the construction becomes a parallelogram. Fixed side lengths do not fix its angle: leaning the figure changes the height and area without changing any side. Perimeter remains the sum of the four supplied lengths.
A trapezoid is a rectangle of its average width
The midsegment, the line joining the midpoints of the two legs, is the average of the parallel sides. For bases of 8 and 12 that average is 10, and multiplying it by a height of 5 gives an area of 50, which is the same answer the half-sum formula produces because they are one calculation written two ways.
That is worth holding on to, because it makes the formula unnecessary to memorise. Average the two parallel sides, then treat the shape as a rectangle of that width. It is exact rather than an approximation, and it holds for every trapezoid however far the legs lean.
Reading it that way also shows why the formula needs no special cases. Let the shorter parallel side shrink to zero and the shape becomes a triangle: half of 8 plus 0, at a height of 5, gives 20, which is exactly half base times height. Let the two sides become equal instead and it becomes a rectangle, with bases of 10 and 10 at a height of 6 giving 60, which is simply 10 times 6.
Trapezoid and trapezium terminology
American and British terminology reverses these names. The American trapezoid handled here has parallel opposite sides and is commonly called a trapezium in Britain. In British usage, a trapezoid instead has no parallel sides and does not fit this calculator’s formulas.
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The formula, worked line by line
Let B and b be the parallel side lengths, h their perpendicular separation, A the area, and c and d the two legs. The basic area formula is the average base length multiplied by height.
For the four-side mode, take B as the longer parallel side and set Δ = B − b. A leg’s horizontal projection can then be found from the two leg lengths and Δ.
A = (B + b)h/2
h = 2A/(B + b)
B = 2A/h − b
b = 2A/h − B
P = B + b + c + d
Δ = B − b
x = (Δ² + c² − d²)/(2Δ)
h = √(c² − x²)The projection formula requires a non-zero base difference. When Δ = 0, fixed side lengths describe a parallelogram whose height varies with its angle, so area is underdetermined.
Perimeter is independent of height but requires every boundary side. Supplying only the parallel sides and height leaves the two leg lengths unknown, so no perimeter is reported.
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Questions people ask
Is a slanted leg the same as the height?
No. Height is the perpendicular distance between the parallel sides. A genuinely slanted leg is longer than that perpendicular component, so using it in the area formula produces an area that is too large.
Can perimeter be found from two bases and height?
No. Those measurements determine the area, but the legs can change while the bases and perpendicular height remain fixed. Perimeter is reported only when both leg lengths were supplied, and the result includes an explicit availability flag.
Why are equal parallel sides underdetermined in four-side mode?
Equal opposite parallel sides make the figure a parallelogram. Its fixed sides can lean through different angles, changing the perpendicular height and area without changing any side length. The perimeter remains determined because all four sides are known.
What is the difference between underdetermined and impossible?
Underdetermined means the supplied measurements describe more than one valid geometry with different requested results. Impossible means the measurements cannot form the required figure. Equal bases in four-side mode are underdetermined, not inherently impossible.
Is a trapezoid the same as a trapezium?
The terminology changes by region. This calculator uses trapezoid in the American sense for a quadrilateral with parallel opposite sides. British usage calls that figure a trapezium and uses trapezoid for a quadrilateral with no parallel sides.
Sources
Where the constants and formulas on this page come from. Each line names the figure it backs.
Supports the area relationship based on the parallel sides and perpendicular height.
Trapezoid — Wolfram MathWorld
Supports treating the bases, legs and perpendicular height as length measurements.
SI Units – Length — National Institute of Standards and Technology
Supports expressing trapezoid area with a squared unit derived from the selected length unit.
Guide for the Use of the International System of Units (SI), NIST SP 811 — National Institute of Standards and Technology
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