The area of any quadrilateral (convex, concave, crossed) is equal to half the product of the diagonals multiplied by the sine of the angle between them. As shown in the dynamic sketch below the area ABCD = ½AC x BD x sin ∠DEA.
It's surprising though that this simple formula appears to be completely absent from high school textbooks.
The reader/viewer is requested to investigate the validity of the result by dragging any of the red vertices A, B, C or D.
Challenge
1) Can you explain why (prove that) the above result is true?
2) Drag the vertices so that ABCD becomes concave or crossed. Can you also explain why (prove) that the result remains true in these cases?
Note: When the diagonals are perpendicular to each other, the above formula obviously reduces to the Area Formula of an Orthodiagonal quadrilateral.
Reference
Harries, J. (2002). Area of a Quadrilateral. The Mathematical Gazette, Vol. 86, No. 506 (Jul.), pp. 310-311. DOI: https://doi.org/10.2307/3621873.
Related Links
Finding the Area of a Crossed Quadrilateral (PDF)
Crossed Quadrilateral Properties
The Area of an Orthodiagonal quadrilateral
International Mathematical Talent Search (IMTS) Area Ratio Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
Another parallelogram area ratio
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof
Areas Ratios (Rethinking Proof activity)
Varignon Parallelogram Area (Rethinking Proof activity, incl. generalization to area crossed quad)
Area ratios of some polygons inscribed in quadrilaterals and triangles (Generalization of Varignon's theorem)
An Area Preserving Transformation: Shearing
A Surprising Constant Area Sum Involving Translating Figures
Sylvie's Theorem
Some Parallelo-hexagon Area Ratios
Feynman's Triangle: Some Generalizations & Variations
Feynman Triangle & Parallelogram Variations
Feynman Parallelogram Generalization
The theorem of Hippocrates (470 – c. 410 BC)
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
External Links
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on Various Topics
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Michael de Villiers, created with WebSketchpad, 29 January 2020; updated 3 April 2024; 16 March 2026.