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.
Motivation: triangles
The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, c
s = a + b + c 2 . {\displaystyle s={\frac {a+b+c}{2}}.}
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 A, B, B', C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (AA', BB', CC', shown in red in the diagram) are known as splitters, and
s = | A B | + | A ′ B | = | A B | + | A B ′ | = | A C | + | A ′ C | = | A C | + | A C ′ | = | B C | + | B ′ C | = | B C | + | B C ′ | . {\displaystyle {\begin{aligned}s&=|AB|+|A'B|=|AB|+|AB'|=|AC|+|A'C|\\&=|AC|+|AC'|=|BC|+|B'C|=|BC|+|BC'|.\end{aligned}}}
The three splitters concur at the Nagel point of the triangle. A 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 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 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 A of any triangle is the product of its inradius (the radius of its inscribed circle) and its semiperimeter:
A = r s . {\displaystyle A=rs.}
The area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Heron's formula:
A = s ( s − a ) ( s − b ) ( s − c ) . {\displaystyle A={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}.}
The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths:
R = a b c 4 s ( s − a ) ( s − b ) ( s − c ) . {\displaystyle R={\frac {abc}{4{\sqrt {s(s-a)(s-b)(s-c)}}}}.}
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