Mathematics · Geometry · Plane Geometry
Trapezoid Area Calculator
Calculates the area of a trapezoid given its two parallel bases and height using the standard geometric formula.
Calculator
Formula
A is the area of the trapezoid; a and b are the lengths of the two parallel sides (bases); h is the perpendicular height between the two parallel bases.
Source: Euclid's Elements, Book I; standard plane geometry reference (e.g., NIST Digital Library of Mathematical Functions).
How it works
A trapezoid is defined as a flat (planar) four-sided polygon in which exactly two sides are parallel to each other. These parallel sides are called the bases, conventionally labeled a (the shorter base) and b (the longer base). The height (h) is the perpendicular distance between the two bases — not the length of any slanted leg. The two non-parallel sides are called the legs and do not appear in the area formula.
The area formula is derived by noting that a trapezoid can be split into a rectangle and one or two right triangles, or equivalently by observing that the trapezoid is exactly half the area of a parallelogram with base equal to the sum of the two parallel sides. This gives the elegant result: A = ½ × (a + b) × h. The term (a + b)/2 is the arithmetic mean of the two bases, also called the median of the trapezoid — the segment connecting the midpoints of the two legs. The area equals the median length multiplied by the height, which provides an intuitive geometric interpretation: the trapezoid behaves like a rectangle whose width is the average of its two bases.
Practical applications of this formula are widespread. Surveyors use it to estimate land areas with one sloped boundary, architects use it to calculate the area of trapezoidal roof sections and facades, and civil engineers apply it to cross-sectional analysis of trapezoidal drainage channels. In numerical mathematics, the trapezoid formula is also the basis of the trapezoidal rule for numerical integration, where curved areas are approximated by summing many thin trapezoidal slices.
Worked example
Problem: A plot of land has two parallel boundary fences of lengths a = 12 m and b = 20 m, with a perpendicular distance of h = 8 m between them. What is the area of the plot?
Step 1 — Identify the inputs: Shorter base a = 12 m, longer base b = 20 m, height h = 8 m.
Step 2 — Sum the bases: a + b = 12 + 20 = 32 m.
Step 3 — Apply the formula: A = ½ × (a + b) × h = 0.5 × 32 × 8 = 128 m².
Interpretation: The land plot covers an area of 128 square metres. Note that this is identical to the area of a rectangle measuring 16 m × 8 m, confirming that the median (16 m) acts as the effective width.
Special case — Parallelogram check: If both bases were equal, say a = b = 16 m, the formula gives A = ½ × 32 × 8 = 128 m² as well, consistent with the rectangle/parallelogram formula A = base × height.
Limitations & notes
Height must be perpendicular: The formula requires the true perpendicular height between the parallel bases, not the slant length of the leg. A common mistake is measuring along the inclined leg rather than dropping a true perpendicular. If only leg lengths and base lengths are known, the height must first be derived using the Pythagorean theorem or trigonometry before applying this formula.
Positive dimensions only: Both bases and the height must be strictly positive real numbers. A base of zero degenerates the trapezoid into a triangle, and negative or zero height is geometrically meaningless. The calculator will return a nonsensical result if non-positive values are entered.
Parallelism assumption: The formula assumes exactly two sides are parallel. If neither pair of sides is truly parallel (a general quadrilateral), the trapezoid formula will give an incorrect answer. For irregular quadrilaterals, the area must be calculated by dividing the shape into triangles.
Unit consistency: All three inputs must be expressed in the same unit. Mixing metres and centimetres, for example, will produce a numerically incorrect result. Convert all measurements to a single unit before entering them.
2D plane figures only: This calculator applies to flat, planar trapezoids. It does not account for curved surfaces, warped quadrilaterals, or three-dimensional prismatic volumes (a trapezoidal prism requires multiplying the cross-sectional area by the prism's depth).
Frequently asked questions
What is the difference between a trapezoid and a trapezium?
The terms refer to the same shape but follow different regional conventions. In North America, a trapezoid is a quadrilateral with exactly one pair of parallel sides, while a trapezium has no parallel sides. In the United Kingdom and much of the rest of the world, the names are reversed. This calculator uses the North American convention, but the formula A = ½(a + b)h applies universally regardless of which term is used.
How do I find the height of a trapezoid if only the side lengths are known?
If you know both bases (a and b) and both leg lengths (c and d), you can calculate the height using the Pythagorean theorem. For a right trapezoid, one leg is already perpendicular and equals h directly. For an isosceles trapezoid with equal legs of length c, the height is h = √(c² − ((b − a)/2)²). For a general trapezoid, decompose it into a rectangle and triangles, solve for the height, then apply the area formula.
Does this formula work for a rectangle or parallelogram?
Yes. A rectangle and a parallelogram are special cases of a trapezoid where both pairs of opposite sides are parallel. When a = b (both bases equal), the formula simplifies to A = ½ × 2a × h = a × h, which is exactly the rectangle or parallelogram area formula. This confirms that the trapezoid formula is a generalisation of those more familiar formulas.
What is the median of a trapezoid and how does it relate to the area?
The median (also called the midsegment) of a trapezoid is the line segment connecting the midpoints of the two non-parallel legs. Its length equals the arithmetic mean of the two bases: m = (a + b)/2. The area of the trapezoid equals the median multiplied by the height: A = m × h. This provides a clean geometric interpretation — the trapezoid has the same area as a rectangle with width equal to the median and height equal to h.
How is the trapezoid area formula used in numerical integration?
The trapezoidal rule in calculus approximates the definite integral of a function by dividing the area under a curve into thin vertical trapezoids. Each strip has parallel vertical sides of heights f(xᵢ) and f(xᵢ₊₁) and a width (base) of Δx. Summing the areas of all strips — each computed with A = ½(f(xᵢ) + f(xᵢ₊₁)) × Δx — gives the total approximate integral. The accuracy improves as the strip width Δx decreases.
Last updated: 2025-01-15 · Formula verified against primary sources.