Area Quadrilateral Calculator

Background.

Ask for the area of a triangle with sides 3, 4, 5 and one answer exists: 6. Ask for the area of a quadrilateral with sides 3, 4, 3, 4 and the honest reply is a question — which one? Those four sides bound a rectangle of area 12, every slimmer parallelogram down to a degenerate sliver of area 0, and kites besides: quadrilaterals hinge. Four sides fix a triangle but not a four-sided shape, and any calculator that returns an area from four sides alone has silently guessed the shape for you.

The missing fact is one diagonal. A diagonal braces the hinge — carpenters know this as the reason a gate gets a cross-brace — and mathematically it splits the quadrilateral into two triangles, each now rigid with three known sides. Heron's classical formula turns each triangle's three sides into its area, and the quadrilateral's area is their sum. That is exactly this page's method: enter sides a, b, c, d and the diagonal that separates the (a, b) corner-pair from the (c, d) pair, and it returns the summed area plus the perimeter.

The approach handles any convex quadrilateral — no need for right angles, parallel sides, or a recognised name — which makes it the practical tool for irregular plots of land, sheet-metal offcuts, and room floors that refuse to be rectangles. Field surveyors have measured exactly these five lengths with a tape for centuries precisely because they determine the figure.

Two cautions frame the tool: the diagonal must actually reach between the correct opposite corners (each triangle's three sides must satisfy the triangle inequality, which the page checks), and for a non-convex, dented quadrilateral the split is only valid across a diagonal that stays inside the shape — as the scope note beside the result records.

What is area quadrilateral calculator?

This calculator finds the area of a general quadrilateral from its four sides and one diagonal, by triangulation: the diagonal e splits the figure into a triangle with sides (a, b, e) and another with sides (c, d, e), each of whose areas follows from Heron's formula — √(s(s−x)(s−y)(s−z)) with s the semi-perimeter — and the total is their sum. The diagonal is not optional decoration: four side lengths leave a quadrilateral free to flex through a whole family of shapes with different areas, and the diagonal is the brace that pins down which shape you mean. Perimeter is the plain sum a+b+c+d.

How to use this calculator.

1. 01Label the sides in order around the shape — a, b, c, d — and measure the diagonal that runs between the corner where a meets d and the corner where b meets c, so it separates the a-b pair from the c-d pair.
2. 02Enter all five lengths in the same unit; the area returns in that unit squared.
3. 03If the calculator rejects the inputs, the three sides of one triangle violate the triangle inequality — almost always a sign the other diagonal was measured, or a side order was swapped.
4. 04For real plots and rooms, measure the diagonal rather than deriving it: a taped diagonal is the ground truth that makes the five numbers determine the figure.
5. 05Cross-check irregular shapes by computing with the other diagonal too (relabelling sides accordingly) — agreement of the two areas is a strong audit of all five measurements.

The formula.

Triangulation rests on rigidity: three sides determine a triangle completely (SSS congruence), so a quadrilateral cut by a diagonal becomes two solvable pieces. Heron's formula supplies each piece's area from sides alone — for sides x, y, z and semi-perimeter s = (x+y+z)/2, area = √(s(s−x)(s−y)(s−z)) — no angles or heights required, which is what makes tape-measure-only surveying possible. The quadrilateral's area is Heron(a, b, e) + Heron(c, d, e) for diagonal e, and its correctness needs the diagonal to lie inside the figure (true for every diagonal of a convex quadrilateral; true for one of the two in a concave one). The negative test is as informative as the positive: if s(s−x)(s−y)(s−z) goes negative, no triangle with those sides exists — the entered ‘diagonal’ cannot span those corners — and the page refuses rather than inventing a shape. The engine evaluates both Heron roots and the sum in Decimal arithmetic, rounding once to twelve significant digits.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06. independence: primary; access: open.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06. independence: secondary-check; access: open.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06. independence: primary; access: open.