Semiperimeter

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 "s".

The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths "a", "b", and "c" is:$s\; =\; frac\{a+b+c\}\{2\}.$

The area of any triangle is the product of its inradius and its semiperimeter; the same area formula also applies to tangential quadrilaterals, in which pairs of opposite sides have lengths adding to the semiperimeter. The area of a triangle can also be calculated from its semiperimeter and side lengths using Heron's formula:

:$ext\{area\}\; =\; sqrt\{sleft(s-a\; ight)left(s-b\; ight)left(s-c\; ight)\}.$

The simplest form of Brahmagupta's formula, for the area of a cyclic quadrilateral, has a similar form:

:$ext\{area\}\; =\; sqrt\{left(s-a\; ight)left(s-b\; ight)left(s-c\; ight)left(s-d\; ight)\}.$

The circumradius "R" of a triangle can also be calculated from the semiperimeter and side lengths::$2R\; =\; frac\{abc\}\; \{2sqrt\{s(s-a)(s-b)(s-c).$This formula can be derived from the law of sines.

The radius of the incircle (also known as the inradius) is

: $sqrt\{frac\{(s-a)(s-b)(s-c)\}\{s.$

In any triangle, the points where the excircles touch the triangle and the opposite vertices of the triangle partition the triangle's perimeter into two equal lengths. That is, if A, B, C, A', B', and C' are as shown in the figure, then

:$s\; =\; |AB|+|A\text{'}B|=|AB|+|AB\text{'}|=|AC|+|A\text{'}C|=|AC|+|AC\text{'}|=|BC|+|B\text{'}C|=|BC|+|BC\text{'}|.$

If one connects each such point of tangency with its opposite vertex by a line (shown red in the figure), these three lines meet in the Nagel point of the triangle.

External links

* [http://agutie.homestead.com/files/semiperimeterexcirclesinc1.htm Semiperimeter, incircle and excircles of a triangle] by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".

*mathworld | title = Semiperimeter | urlname = Semiperimeter