Trapezoid area calculator

Definition

A trapezoid is a quadrilateral with one pair of parallel sides called the bases. Denote the base lengths by a and b and the perpendicular distance between the lines containing the bases by height h. On the Euclidean plane the area is A = ((a + b)/2)·h. This page implements exactly A = (a + b) · h / 2 from positive a, b, h entered in the form fields.

Height is not the length of a slanted leg. A leg joins the ends of the bases and is usually longer than h; substituting a leg for h overstates the area. The area unit is the square of the length unit (cm² or in² from the unit switch).

The primary result is area A. The panel also shows a, b, the mean (a+b)/2, height h, the sum a+b, and |a−b|.

Historical and conceptual context

The trapezoid and the mean-base area formula belong to the Euclidean school canon. In Euclid’s Elements (Book I), areas of rectilinear figures are reduced to parallelograms and triangles of equal height; a trapezoid sits between a triangle (one base tending to zero) and a parallelogram (a = b). MathWorld (Trapezoid) and Wikipedia (Trapezoid) record A = ½(a+b)h and the midsegment equal to (a+b)/2.

In US school usage a trapezoid usually has exactly one pair of parallel sides; British “trapezium” terminology can differ. This page computes the two-parallel-bases model with perpendicular height — isosceles symmetry is not required.

In practice the formula appears in channel cross-sections, roof profiles, trapezoidal panels, and area estimates when two parallel edges and the height are known. Neighboring rectangle and parallelogram calculators take one horizontal length and a height without a second base.

Formal properties

Domain: a > 0, b > 0, h > 0. Codomain: A > 0. The map (a,b,h) ↦ A is linear in each argument separately when the others are fixed, and homogeneous of degree 2 when all three lengths scale together.

Identities: A = ((a+b)/2)·h = (a+b)·h/2; midsegment m = (a+b)/2; A = m·h. When a = b the formula reduces to A = a·h (parallelogram / rectangle at right angles). As b → 0⁺ one obtains A → (a/2)·h — a triangle of base a and height h.

Commutativity: A(a,b,h) = A(b,a,h). The difference |a−b| is not needed in the area formula itself but describes the overhang and appears in leg projections.

Decomposition: if a ≥ b, trapezoid = rectangle of base b and height h plus two triangles of combined base (a−b) and height h, hence A = b·h + ½(a−b)·h = ((a+b)/2)·h. The identity confirms the formula without integration.

In vector form, the area of a quadrilateral with vertices on parallel lines can be written via a cross product; with bases parallel to an axis the result reduces to mean length times distance between the lines.

Full derivation

Step 1. Let AB and DC be the bases of lengths a and b (AB ∥ DC), and let h be the distance between lines AB and DC.

Step 2. Drop perpendiculars from D and C onto AB. This yields a rectangle (or height-aligned parallelogram) on the shorter base and two right triangles whose bases sum to |a−b|.

Step 3. Area = shorter·h + ½|a−b|·h = ((a+b)/2)·h after algebraic reduction.

Step 4 (midsegment). The segment joining the midpoints of the legs has length (a+b)/2; the trapezoid has the same area as a rectangle of that base and height h.

Step 5. Equivalently A = (a+b)·h/2 — the same arithmetic the calculator performs.

Special cases

a = 10, b = 6, h = 4: mean 8, A = 32. a = 8, b = 4, h = 5: A = 30. a = 12, b = 8, h = 3: A = 30. a = b = 5, h = 5: A = 25 (parallelogram). a = 15, b = 5, h = 4: A = 40. a = 20, b = 10, h = 6: A = 90. a = 7, b = 3, h = 2: A = 10. a = 9, b = 6, h = 8: A = 60.

As h → 0⁺, A → 0. Doubling both bases and the height multiplies area by eight (degree 2). When a = b the figure is a parallelogram; the page still uses the same formula.

A blank second base is not treated as a triangle — three positive lengths are required.

Numerical and computational notes

The calculator computes A = ((a + b) * h) / 2 in floating-point arithmetic and formats with the page locale (pl-PL / en-US). Non-positive or non-numeric inputs are rejected. Comma and dot decimals are equivalent.

Examples fill the sets from the Examples section and refresh the SVG diagram. Drawing scale fits the longer base to the view width; it does not change the numeric result.

cm/in labels depend only on the header units switch. The calculator does not sort the longer base — field order is free.

Comparison with rectangle a×h or a×b is intentionally different: a trapezoid always mixes both bases through their mean. At a = 10, b = 6, h = 4 a 10×6 rectangle would be 60, while the trapezoid is 32.

Related notions and sources

Related: rectangle area a·b, parallelogram area a·h, triangle area ½a·h (limit b → 0), trapezoid midsegment, Pythagoras for a leg.

Sources: MathWorld — Trapezoid; Wikipedia — Trapezoid; Euclid, Elements, Book I (areas via parallelograms and triangles); standard school texts — A = ((a+b)/2)·h. The calculator computes A = ((a+b)/2)·h.