geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the semiperimeter of a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

is half its

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for

triangles A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensiona ...

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 .

Motivation: triangles

The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths :

s = \frac.

Properties

In any triangle, any vertex and the point where the opposite

excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

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 are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (, shown in red in the diagram) are known as splitters, and :

\begin
s &= , AB, +, A'B, =, AB, +, AB', =, AC, +, A'C,  \\
  &= , AC, +, AC', =, BC, +, B'C, =, BC, +, BC', .
\end

The three splitters concur at the Nagel point of the triangle. A

cleaver A cleaver is a large knife that varies in its shape but usually resembles a rectangular-bladed tomahawk. It is largely used as a kitchen knife, kitchen or butcher knife and is mostly intended for splitting up large pieces of soft bones and slas ...

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 In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...

of the medial triangle; the Spieker center is the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

of all the points on the triangle's edges. A line through the triangle's

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...

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 In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...

, the longest side length of a triangle is less than the semiperimeter.

Formulas involving the semiperimeter

For triangles

The area of any triangle is the product of its

inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

(the radius of its inscribed circle) and its semiperimeter: :

A = rs.

The area of a triangle can also be calculated from its semiperimeter and side lengths using

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...

: :

A = \sqrt.

The circumradius of a triangle can also be calculated from the semiperimeter and side lengths: :

R = \frac .

This formula can be derived from the

law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...

. The inradius is :

r = \sqrt.

The

law of cotangents In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. Just as three quantities whose equality is expressed by the law of sines are equal to t ...

gives the

cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

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 internal bisector of the angle opposite the side of length is :

t_a= \frac.

In a

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

, the radius of the

on the

hypotenuse In geometry, a hypotenuse is the side of a right triangle opposite to the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called '' catheti'' or ''legs''. Every rectangle can be divided ...

equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is

(s-a)(s-b)

where are the legs.

For quadrilaterals

The formula for the semiperimeter of a

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

with side lengths is :

s = \frac.

One of the triangle area formulas involving the semiperimeter also applies to

tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This cir ...

s, which have an incircle and in which (according to Pitot's theorem) pairs of opposite sides have lengths summing to the semiperimeter—namely, the area is the product of the inradius and the semiperimeter: :

K = rs.

The simplest form of Brahmagupta's formula for the area of a

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

has a form similar to that of Heron's formula for the triangle area: :

K = \sqrt.

Bretschneider's formula In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not. The formula also works on crossed quadrilaterals provided ...

generalizes this to all

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

quadrilaterals: :

K = \sqrt ,

in which and are two opposite angles. The four sides of a

bicentric quadrilateral In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' ...

are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius.

Regular polygons

The area of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

is the product of its semiperimeter and its

apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...

Circles

The semiperimeter of a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, also called the semi

circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...

, is directly proportional to its

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

: :

s = \pi \cdot r.\!

The constant of proportionality is the number pi, .

References

External links

*{{mathworld , title = Semiperimeter , urlname = Semiperimeter Triangle geometry