Area
The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula
where s, the semiperimeter, is . It is a corollary to Bretschneider's formula since opposite angles are supplementary. If also d = 0, the cyclic quadrilateral becomes a triangle and the formula is reduced to Heron's formula.
The cyclic quadrilateral has maximal area among all quadrilaterals having the same sequence of side lengths. This is another corollary to Bretschneider's formula. It can also be proved using calculus.
Four unequal lengths, each less than then sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area. Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.
The area of a cyclic quadrilateral with successive sides a, b, c, d and angle B between sides a and b can be expressed as
or
where θ is the angle between the diagonals. Provided A is not a right angle, the area can also be expressed as
Another formula is
where R is the radius in the circumcircle. As a direct consequence,
where there is equality if and only if the quadrilateral is a square.
Read more about this topic: Cyclic Quadrilateral
Famous quotes containing the word area:
“Whatever an artist’s personal feelings are, as soon as an artist fills a certain area on the canvas or circumscribes it, he becomes historical. He acts from or upon other artists.”
—Willem De Kooning (b. 1904)
“I am aware of the damp souls of housemaids
Sprouting despondently at area gates.”
—T.S. (Thomas Stearns)
“Now for civil service reform. Legislation must be prepared and executive rules and maxims. We must limit and narrow the area of patronage. We must diminish the evils of office-seeking. We must stop interference of federal officers with elections. We must be relieved of congressional dictation as to appointments.”
—Rutherford Birchard Hayes (1822–1893)