Editing Semiperimeter (section)

=== For quadrilaterals ===
The formula for the semiperimeter of a [[quadrilateral]] with side lengths {{mvar|a, b, c, d}} is
:<math>s = \frac{a+b+c+d}{2}.</math>

One of the triangle area formulas involving the semiperimeter also applies to [[tangential quadrilateral]]s, which have an incircle and in which (according to [[Pitot's theorem]]) pairs of opposite sides have lengths summing to the semiperimeter&mdash;namely, the area is the product of the inradius and the semiperimeter:

:<math> K = rs.</math>

The simplest form of [[Brahmagupta's formula]] for the area of a [[cyclic quadrilateral]] has a form similar to that of Heron's formula for the triangle area:

:<math>K = \sqrt{\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)}.</math>

[[Bretschneider's formula]] generalizes this to all [[convex polygon|convex]] quadrilaterals:

:<math> K = \sqrt {(s-a)(s-b)(s-c)(s-d) - abcd  \cdot \cos^2 \left(\frac{\alpha + \gamma}{2}\right)},</math>

in which {{mvar|α}} and {{mvar|γ}} are two opposite angles.

The four sides of a [[bicentric quadrilateral]] are the four solutions of [[Bicentric quadrilateral#Inradius and circumradius|a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius]].