Area of a quadrilateral from the lengths of its sides
Last week Heron's formula came up in the post An Unexpected Triangle. Given the lengths of the sides of a triangle, there is a simple expression for the area of the triangle.
where the sides are a, b, and c and s is the semiperimeter, half the perimeter.
Is there an analogous formula for the area of a quadrilateral? Yes and no. If the quadrilateral is cyclic, meaning there exists a circle going through all four of its vertices, then Brahmagupta's formula for the area of a quadrilateral is a direct generalization of Heron's formula for the area of a triangle. If the sides of the cyclic quadrilateral are a, b, c, and d, then the area of the quadrilateral is
where again s is the semiperimeter.
But in general, the area of a quadrilateral is not determined by the length of its sides alone. There is a more general expression, Bretschneider's formula, that expresses the area of a general quadrilateral in terms of the lengths of its sides and the sum of two opposite angles. (Either pair of opposite angles lead to the same value.)
In a cyclic quadrilateral, the opposite angles and add up to , and so the cosine term drops out.
The contrast between the triangle and the quadrilateral touches on an area of math called distance geometry. At first this term may sound redundant. Isn't geometry all about distances? Well, no. It is also about angles. Distance geometry seeks results, like Heron's theorem, that only depend on distances.
Related posts- Shoelace formula for polygon area
- Area of a triangle in the complex plane
- Area of a quadrilateral as a determinant