1. Area from bases and height
For a=10 cm, b=6 cm and h=5 cm:
A=(10+6)×5/2=40 cm².
Trapezoid Calculator: Area, Perimeter, Height and Sides
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Reference diagram
Labels identify the two parallel bases, two legs, perpendicular height and diagonals. The drawing is a reference and is not to scale.
Results and calculation steps
Choose a mode and enter the required values. Results will separate supplied measurements from derived properties.
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Trapezoid formula reference
a and b are the parallel bases, c and d are the legs, h is perpendicular height, A is area, P is perimeter and m is the midsegment. Special-shape formulas apply only under the stated assumption.
| Find | Formula | Valid for |
|---|---|---|
| Area | A = (a+b)h/2 | Every trapezoid |
| Height | h = 2A/(a+b) | Every trapezoid when area and bases are known |
| Either base | a = 2A/h−b; b = 2A/h−a | Every trapezoid when area, height and the other base are known |
| Perimeter | P = a+b+c+d | Every trapezoid |
| A missing side from perimeter | a = P−b−c−d, with equivalent rearrangements for b, c or d | Every trapezoid, subject to geometric validity |
| Midsegment | m = (a+b)/2 | Every trapezoid |
| Isosceles leg and height | L = √(h²+((a−b)/2)²); h = √(L²−((a−b)/2)²) | Isosceles only |
| Isosceles diagonal | p = q = √(h²+((a+b)/2)²) | Isosceles only |
| Right trapezoid | one leg = h; other leg = √(h²+(a−b)²) | Right only |
| General height from four sides | s=a−b; x=(c²−d²+s²)/(2s); h=√(c²−x²) | General, unequal bases, supported orientation |
Why the area rearrangements work
Starting with A=(a+b)h/2, multiply by 2 and divide by a+b to get h=2A/(a+b). Dividing by h instead gives a+b=2A/h, so subtract the known base to find the missing one. For four-side geometry, dropping perpendiculars from the top base creates two right triangles; subtracting their Pythagorean equations produces the projection x shown in the table.
Worked examples
2. Height from area and bases
For A=66 cm², a=14 cm and b=8 cm:
h=2×66/(14+8)=6 cm.
3. Isosceles properties
For a=10 cm, b=6 cm and h=3 cm, each leg is √13≈3.61 cm, P≈23.21 cm, and each diagonal is √73≈8.54 cm.
Methodology, accuracy and limitations
Author: Starlight Robotics
Review method: formula derivation plus the normal, special-shape, impossible and underdetermined browser test cases below
Reviewed:
Numerical policy: calculations keep JavaScript floating-point precision; displayed values alone are rounded. Consistency checks use a relative tolerance of 1×10⁻⁹.
References: OpenStax, Contemporary Mathematics §10.6 (area) and NIST SI Units — Length.
Supported general orientation: the bottom base runs left to right and the top base is represented above it as (x,h) to (x+b,h). Acute, obtuse and overhanging top-base positions are supported. The mirror image below the base is equivalent for lengths and angle magnitudes.
Limitations: bases must be positive and a fully solved four-side general case needs unequal bases. Equal bases with arbitrary legs do not uniquely fix the horizontal offset. Area plus bases and height does not determine general legs, diagonals, angles or perimeter.
Verification cases
| Case | Inputs | Expected check |
|---|---|---|
| Normal area | a=10, b=6, h=5 | A=40 |
| Isosceles | a=10, b=6, h=3 | L=√13, P=10+6+2√13, diagonals √73 |
| Right | a=10, b=6, h=3 | legs 3 and 5, P=24 |
| Impossible | a=10, b=6, c=1, d=1 | Rejected: the legs cannot span the base difference |
| Underdetermined | General a=10, b=6, h=5 | A=40; legs, perimeter, diagonals and angles remain unknown |
All calculations run locally in your browser; values are not uploaded.
Frequently asked questions
Which inputs determine a trapezoid?
Use the requirements shown for the selected mode. For example, area needs both bases and height; height needs both bases and area; a general trapezoid needs both bases and both legs. Two arbitrary values usually do not determine a unique trapezoid.
How do I find the height of a trapezoid?
If area A and bases a and b are known, use h = 2A/(a+b). For an isosceles trapezoid with leg L, use h = √(L²−((a−b)/2)²).
How do I find a missing base?
From area and height, use a = 2A/h−b or b = 2A/h−a. From perimeter and the other three sides, subtract those three known sides from P.
Does area determine the legs or perimeter?
No. Bases and height determine area, but a general trapezoid can slide sideways and have different legs, diagonals, angles and perimeter. Select Isosceles or Right only when that assumption is given.
What is the difference between trapezoid and trapezium?
In US usage, trapezoid usually means a quadrilateral with a pair of parallel sides. In UK usage, that shape is usually called a trapezium. This calculator uses trapezoid in the US sense.
What assumptions are made for isosceles and right trapezoids?
Isosceles assumes equal legs and symmetric base overhangs. Right assumes one selected leg is perpendicular to the bases and therefore equals the height. General makes neither assumption.
Why are my dimensions invalid or underdetermined?
Dimensions must be positive and mutually consistent. For two unequal bases and two legs, the leg lengths and absolute base difference must form a non-degenerate triangle. A result is underdetermined when the supplied values allow more than one shape.
Can I enter fractions and use different units?
You can enter decimals, fractions such as 3/4, or mixed numbers such as 2 1/2. Choose one shared unit and enter every length in that unit; the selector labels results but does not convert mixed units.