semiperimeter

Semiperimeter is a geometric term referring to half of a polygon’s perimeter. In general, for any polygon with perimeter P, the semiperimeter s is P/2. The term is most commonly associated with triangles, where s is defined as one-half the sum of the side lengths: s = (a + b + c)/2, for a triangle with side lengths a, b, c.

Heron’s formula expresses the area Δ of a triangle in terms of its semiperimeter: Δ = sqrt[s (s − a) (s − b) (s − c)]. The semiperimeter also appears in formulas relating area, inradius, and side lengths: Δ = r s, where r is the inradius.

Beyond triangles, the semiperimeter applies to any polygon as half of its perimeter. For polygons that admit an incircle (tangential polygons), the area also satisfies Δ = r s, linking the inradius r to s. In the broader case, s is simply half the perimeter P, and is used in various geometric and optimization contexts.

An example: a 3-4-5 triangle has s = (3 + 4 + 5)/2 = 6, area Δ = sqrt(6·3·2·1) = 6, and inradius r = Δ / s = 1. The concept provides a convenient shorthand for expressing relationships among side lengths, area, and inradius in triangles and, more generally, in polygons.