PlanetMath: area of a quadrilateral

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area of a quadrilateral

(Theorem)

Let be the lengths of the sides of a quadrilateral and be its area. Let be the semiperimeter. Then

$\displaystyle K^2 = (s-a)(s-b)(s-c)(s-d) - abcd \cos^2 \left(\frac{\theta+\phi}{2}\right )$

where $\theta$ and $\phi$ are opposite angles of the quadrilateral. Letting $d \to 0$ we obtain Heron's formula for the area of a triangle.

Bibliography

1: C.A. Bretschneider, Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Math. 2, (1842), 225-261.
2: F. Strehlke, Zwei neue Sätze vom ebenen und shpärischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes. Archiv der Math. 2, (1842) 323-326.

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Cross-references: area of a triangle, Heron's formula, semiperimeter, area, quadrilateral, sides, lengths
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This is version 4 of area of a quadrilateral, born on 2007-04-23, modified 2007-04-24.
Object id is 9248, canonical name is AreaOfAQuadrilateral.
Accessed 2807 times total.

Classification:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

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