Editing Semiperimeter

{{Short description|Half of the sum of side lengths of a polygon}}
In [[geometry]], the '''semiperimeter''' of a [[polygon]] is half its [[perimeter]]. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for [[triangles]] and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter {{mvar|s}}.

==Motivation: triangles==
[[Image:Nagel point.svg|thumb|300px|In any triangle, the distance along the boundary of the triangle from a vertex to the point on the opposite edge touched by an [[excircle]] equals the semiperimeter.]]

The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths {{mvar|a, b, c}}
:<math>s = \frac{a+b+c}{2}.</math>

===Properties===

In any triangle, any vertex and the point where the opposite [[excircle]] touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has a length equal to the semiperimeter. If {{mvar|A, B, B', C'}} are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency ({{mvar|{{overline|AA'}}, {{overline|BB'}}, {{overline|CC'}}}}, shown in red in the diagram) are known as [[Splitter (geometry)|splitters]], and

:<math>\begin{align}
s &= |AB|+|A'B|=|AB|+|AB'|=|AC|+|A'C| \\
  &= |AC|+|AC'|=|BC|+|B'C|=|BC|+|BC'|.
\end{align}</math>

The three splitters [[Concurrent lines|concur]] at the [[Nagel point]] of the triangle.

A [[Cleaver (geometry)|cleaver]] of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter. The three cleavers concur at the [[Spieker center|center of the Spieker circle]], which is the [[incircle]] of the [[medial triangle]]; the Spieker center is the [[center of mass]] of all the points on the triangle's edges.

A line through the triangle's [[incenter]] [[Bisection|bisects]] the perimeter if and only if it also bisects the area.

A triangle's semiperimeter equals the perimeter of its [[medial triangle]].

By the [[triangle inequality]], the longest side length of a triangle is less than the semiperimeter.

==Formulas involving the semiperimeter==

=== For triangles ===
The area {{mvar|A}} of any triangle is the product of its [[inradius]] (the radius of its inscribed circle) and its semiperimeter:

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

The area of a triangle can also be calculated from its semiperimeter and side lengths {{mvar|a, b, c}} using [[Heron's formula]]:

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

The [[circumradius]] {{mvar|R}} of a triangle can also be calculated from the semiperimeter and side lengths:
:<math>R = \frac{abc} {4\sqrt{s(s-a)(s-b)(s-c)}}.</math>
This formula can be derived from the [[law of sines]].

The inradius is

: <math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}. </math>

The [[law of cotangents]] gives the [[cotangent]]s of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius.

The length of the [[Bisection#Angle bisector|internal bisector of the angle]] opposite the side of length {{mvar|a}} is<ref name=Johnson>{{cite book|last=Johnson|first=Roger A.|title=Advanced Euclidean Geometry|year=2007|publisher=Dover|location=Mineola, New York|isbn=9780486462370|page=70}}</ref>

:<math>t_a= \frac{2 \sqrt{bcs(s-a)}}{b+c}.</math>

In a [[right triangle]], the radius of the [[excircle]] on the [[hypotenuse]] equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is <math>(s-a)(s-b)</math> where {{mvar|a, b}} are the legs.

=== 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]].

=== Regular polygons ===
The area of a [[Convex polygon|convex]] [[regular polygon]] is the product of its semiperimeter and its [[apothem]].

==Circles==
The semiperimeter of a [[circle]], also called the semi[[circumference]], is directly proportional to its [[radius]] {{math|''r''}}:

:<math>s = \pi \cdot r.\!</math>

The constant of proportionality is the number [[pi]], {{pi}}.

==See also==
*[[Semidiameter]]

==References==
{{Reflist}}

== External links ==
*{{mathworld | title = Semiperimeter | urlname = Semiperimeter}}

[[Category:Triangle geometry]]