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?
Related Links
Finding the Area of a Crossed Quadrilateral
Crossed Quadrilateral Properties
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
International Mathematical Talent Search (IMTS) 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
Area ratios of some polygons inscribed in quadrilaterals and triangles
An Area Preserving Transformation: Shearing
Sylvie's Theorem
Some Parallelo-hexagon Area Ratios
Cyclic Hexagon Alternate Angles Sum Theorem
Free Download of Geometer's Sketchpad
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Michael de Villiers, created with WebSketchpad, 29 January 2020; updated 3 April 2024.