Trapezoid Area Calculator: Bases, Height, Solve for Either

About this calculator

The trapezoid area calculator takes the two parallel sides of a trapezoid (the bases $a$ and $b$ ) and the perpendicular distance between them (the height $h$ ) and returns the area $A = (a + b)\,h / 2$ . Conversely, given the area and two of the other quantities, it solves for the height or for a base.

A trapezoid is a quadrilateral with one pair of parallel sides, and in the real world it is more common than the rectangle: cross-sections of rivers and ditches, the sides of a roof, wedge-shaped plots of land, the stringers of a staircase, the profile of an embankment, the net of a frustum. Any region cut out by two parallel lines has area equal to “the average length of the two parallel sides × the height”.

What it does not do: it neither needs nor accepts the lengths of the two slanted sides (the legs)—the area does not depend on them; it does not judge whether three given numbers can form a specific trapezoid; it does not compute the perimeter (which needs the legs); and it does not convert units.

Formula and variables

The area of a trapezoid is the average of the two parallel sides times the height:

A = \frac{(a + b)\,h}{2} = m\,h,\qquad m = \frac{a + b}{2}

$m$ is the median (or midsegment, the segment joining the midpoints of the legs); it is parallel to the bases and its length is their average. Solving for the other quantities:

h = \frac{2A}{a + b},\qquad a = \frac{2A}{h} - b

Symbol	Meaning	Calculator field	Unit	Range and notes
$a$	Top base (one parallel side)	Top base a	length	$a \ge 0$ ; default 4; $a = 0$ degenerates to a triangle; can be solved from the area
$b$	Bottom base (the other parallel side)	Bottom base b	length	$b > 0$ ; default 8; which side is called top or bottom does not affect the result
$h$	Height: the perpendicular distance between the parallel sides	Height h	length	$h > 0$ ; default 5; not the leg length; can be solved from the area
$m$	Length of the median	(intermediate)	length	$(a + b)/2$
$A$	Area	Area A	length²	Output; can be entered to solve for $h$ or $a$

The three lengths share one unit; the area is in the square of that unit.

How to use this calculator

Open the panelthis page corresponds to “Math tools → Formulas → Trapezoid calculator” (/#/mathtools/formula?calculator=calculator.trapezoid). It opens with top base a = 4 and bottom base b = 8 filled in but height h empty, treated as 0, so the initial area shows 0—enter the height first (the default example is 5).
Find the areaonce all three lengths are filled in, “Area A”, marked “computed”, updates live to at most 8 decimal places with trailing zeros removed.
Solve for the height from the areaclear “Height h”, enter the area in “Area A” and keep a and b; the height switches to “computed”.
Solve for the top base from the areaclear “Top base a”, enter the area, the bottom base and the height; the top base switches to “computed”. To solve for the bottom base, enter the known parallel side in “Top base a”—the two bases are symmetric and the formula is the same.
Measure the height, not the legthe height is the perpendicular distance between the parallel sides. If you have only the leg length $c$ and its angle $\alpha$ to the base, $h = c\sin\alpha$ ; for an isosceles trapezoid you can also use the Pythagorean theorem, $h = \sqrt{c^2 - \big((b-a)/2\big)^2}$ .

Worked examples

All four examples were recomputed by the engine’s trapezoid compute routine; the display convention is at most 8 decimal places with trailing zeros removed.

Example 1: panel defaults 4, 8, 5

Sum of the bases $a + b = 12$ , median $m = 6$ .
$A = 12 \times 5 / 2 = 30$ .

Interface shows: area 30. If this is an isosceles trapezoid, each leg is $\sqrt{5^2 + 2^2} = 5.385$ —but the area does not use it.

Figure 1: Example 1 and its derivation. Two identical trapezoids fit together exactly into a parallelogram

Example 2: decimals—3.5, 6.5, 2

$a + b = 10$ , $m = 5$ .
$A = 10 \times 2 / 2 = 10$ .

Interface shows: area 10. This is the cross-section of a ditch 2 m deep whose banks measure 3.5 m and 6.5 m (in m²).

Example 3: solving for the height from the area

A trapezoidal plot has an area of 30 m² and parallel sides of 4 m and 8 m; how far apart are they? Clear the height, enter area 30, a = 4, b = 8.

$h = 2A/(a + b) = 60/12 = 5$ .

Interface shows: height 5.

Example 4: solving for the top base from the area

Area 48, bottom base 10, height 6; how long is the top base? Clear the top base, enter area 48, b = 10, h = 6.

$a = 2A/h - b = 96/6 - 10 = 16 - 10 = 6$ .

Interface shows: top base 6. Change the area to 10 (everything else unchanged) and the result is $20/6 - 10 = -6.67$ —a negative value means no such trapezoid exists: a triangle with base 10 and height 6 already has area 30, and a trapezoid’s area cannot be smaller.

The inputs of Example 1 can be entered directly into the panel to reproduce it; change the top base to 8 and the area becomes 40 (a parallelogram); change it to 0 and the area becomes 20 (a triangle).

Principles and derivation

Rearrangement: two trapezoids = one parallelogram

Copy the trapezoid, rotate the copy by 180° and attach it along one leg so that the top and bottom bases join end to end. The result is a parallelogram with base $a + b$ and the same height $h$ (Figure 1, right). Its area is base × height $= (a + b)h$ , and the trapezoid is half of it: $A = (a + b)h/2$ .

Decomposition: a diagonal cuts it into two triangles

Cut the trapezoid along a diagonal into two triangles with bases $a$ and $b$ and the same height $h$ (the distance between the bases): $A = \tfrac12 ah + \tfrac12 bh = \tfrac12(a + b)h$ . This view explains why the leg lengths are irrelevant: slide the top base sideways parallel to itself and neither triangle’s base or height changes, so the area is unchanged—shear invariance, the planar version of Cavalieri’s principle.

The median: “straightening” the trapezoid into a rectangle

The segment joining the midpoints of the legs is the median, of length $m = (a + b)/2$ . Cut off the small triangles at the two corners of the upper half along the median’s height, flip them over and they fit exactly against the two sides of the lower half, turning the trapezoid into an $m \times h$ rectangle. Hence $A = m h$ : the area of a trapezoid is “average width × height”. This is also the intuition for estimating the area of an irregular plot—measure a few widths, average them and multiply by the length.

The trapezoidal rule: approximating the area under a curve

Slice the region under a curve into narrow strips of width $\Delta x$ , approximate each strip by a trapezoid (whose parallel sides are the function values at the ends) and add them up; that is the trapezoidal rule:

\int_{x_0}^{x_n} f(x)\,dx \approx \Delta x \left[\tfrac12 f(x_0) + f(x_1) + \cdots + f(x_{n-1}) + \tfrac12 f(x_n)\right]

Numerical integration in engineering and science, and estimating totals from discrete measurements (water depth, flow speed, power), are all repeated applications of $(a + b)h/2$ . The error is proportional to $\Delta x^2$ and to the second derivative of the function: the more curved the graph and the larger the step, the bigger the gap between trapezoid and curve.

Trapezoids and other quadrilaterals

Rectangles, squares and parallelograms all have a pair of parallel sides. Under the inclusive definition (used in British and American textbooks and by this calculator) they are special cases of the trapezoid and the formula applies automatically (the limiting cases in the examples); under the exclusive definition (exactly one pair of parallel sides, as in mainland Chinese textbooks) they are not trapezoids, but the area formula still holds—the formula does not care about naming. Note also the terminology: in American English trapezoid is the shape with one pair of parallel sides, whereas in British English that shape is a trapezium and trapezoid means a quadrilateral with no parallel sides.

Isosceles trapezoids and leg length

A trapezoid with equal legs is isosceles. Its leg, its height and half the difference of the bases form a right triangle: $c^2 = h^2 + \big((b - a)/2\big)^2$ , so the height follows from the leg (see step 5 of How to use) and the leg from the height. A general trapezoid has unequal legs and needs each leg’s angle to find the height.

Assumptions

$a$ and $b$ are parallel: the formula holds only for the two parallel sides; averaging two non-parallel sides is meaningless.
$h$ is the perpendicular distance: the height must be perpendicular to the bases; entering a leg length overstates the area (leg ≥ height).
Plane figure: the trapezoid lies in one plane; a “skew quadrilateral” in space does not qualify.
One unit: the three lengths share a unit; the area is its square.
Non-negative lengths: the engine does not block negative numbers, but negative lengths have no geometric meaning; a negative solved top base means the given area is too small.
All three fields must be filled in: the panel pre-fills only a and b and treats the empty height as 0; when solving for a with the height empty, the engine uses 1 and the result is meaningless.

Scope and limitations

Can calculate: the area of any trapezoid (including the degenerate parallelogram, rectangle and triangle cases); the height or one parallel side from the area. Results to at most 8 decimal places.
Approximate only: with measurement errors, the area error is about $\Delta A \approx \tfrac{h}{2}(\Delta a + \Delta b) + \tfrac{a+b}{2}\Delta h$ .
Cannot calculate: the perimeter (needs the legs), the area directly from leg lengths (find the height first), irregular quadrilaterals (split into triangles or use coordinates), or the volume of a 3D frustum (use the cylinder / cone or frustum formulas).
Does not: judge whether the given lengths can form a trapezoid, solve for the bottom base (symmetric—enter the known base as “Top base”), or convert units.
Numerics: when $a + b = 0$ , solving for the height divides by zero and shows ∞.

Common mistakes

Using the leg as the height: the height is the perpendicular distance between the bases. For a trapezoid with leg 5.385 and height 5, using the leg inflates the area from 30 to 32.31.
Forgetting to halve: $(a + b)h$ is the area of two trapezoids (a parallelogram).
Multiplying only one base by the height: $bh$ is a parallelogram; a trapezoid needs the average of both bases.
Mixing units: bases in metres and height in centimetres gives a result off by a factor of 100.
Reading the area straight after opening the panel: the height is initially empty and the area 0 is not a result. Enter the height first.
Applying the formula to non-parallel sides: a general quadrilateral has no such simple area formula; confirm which two sides are parallel first.
Accepting a negative solved value as an answer: a negative top base means the area contradicts the other two quantities; check the inputs.

Typical use cases

Teaching: from rectangle to trapezoid to triangle

Hold b = 8 and h = 5 fixed and reduce a step by step from 8 to 0: the area falls linearly from 40 (a parallelogram) to 20 (a triangle), and every step in between is a trapezoid. An intuitive feel for “area = average width × height”.

Cross-sections of ditches, channels and embankments

The trapezoid is the most common channel cross-section. A section with top width 3.5 m, bottom width 6.5 m (or the reverse) and depth 2 m has area 10 m²; multiplied by the flow speed it gives the discharge. The cross-sectional area of an embankment times its length gives the earthwork volume.

Plots and roofs

For a plot with two parallel sides and two that are not, measure the two parallel sides and the distance between them to get the area; the trapezoidal panels on the sides of a pitched roof work the same way, for estimating tiles or sheeting.

Approximate integration of data

For power, flow speed or velocity recorded over time, approximate the “area” between adjacent points by a trapezoid and accumulate to get total energy, volume or displacement—the most everyday form of the trapezoidal rule.

FAQ

Does it matter which side is the top base and which the bottom?

No. The formula is symmetric in the two parallel sides; swapping a and b leaves the result unchanged. When solving, enter the known side as “Top base a” to solve for the other.

Can I find the area from the four sides alone?

Not with this calculator. A trapezoid is determined uniquely by its four sides (given both bases and both legs the height can be solved), but the height must first be found geometrically: split the trapezoid into a parallelogram and a triangle, use Heron’s formula for the triangle’s area and divide by its base to get the height.

Why is the area 0 when I open the panel?

The height field is empty and treated as 0. Enter the height.

Can I use this calculator for parallelograms and triangles?

Yes. With a = b you get the parallelogram area bh; with a = 0 you get the triangle area bh/2.

What does a negative solved top base mean?

The given area is smaller than that of a triangle with the same base and height, so no such trapezoid exists. Check the area, bottom base and height.

How is the trapezoidal rule related to this formula?

The trapezoidal rule slices the region under a curve into many narrow trapezoids, finds each area with $(a + b)h/2$ and adds them up, where $a$ and $b$ are the function values at adjacent points and $h$ is the step size.

References and further reading

The trapezoid definition in the CalcX engine source src/data/formulas.ts ( $A = (a+b)h/2$ , with h and a solved from area); every figure on this page was recomputed by that engine.
Wikipedia, Trapezoid （访问日期：2026-09-08）—the definitional debate (inclusive/exclusive, British/American usage), area, median and other properties.
Wikipedia, Trapezoidal rule （访问日期：2026-09-08）—the trapezoidal rule and its error analysis.
Wolfram MathWorld, Trapezoid （访问日期：2026-09-08）—area, median and diagonal formulas.
Stewart, J. Calculus: Early Transcendentals, 8th ed. Cengage, 2016. §7.7 “Approximate Integration”—derivation and error bound of the trapezoidal rule.

Privacy

All inputs and calculations run inside your browser and are never uploaded to a server. The text on this page is static content; it neither contains nor records any user input.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

Wikipedia: Trapezoid （访问日期：2026-09-08）
Wikipedia: Trapezoidal rule （访问日期：2026-09-08）
Wolfram MathWorld: Trapezoid （访问日期：2026-09-08）
Stewart, J., Calculus: Early Transcendentals, 8th ed., Cengage, 2016, §7.7 “Approximate Integration”

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