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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''.


Triangles

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.


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, C, A', B', and C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (AA', BB', and CC', shown in red in the diagram) are known as splitters, and s = , AB, +, A'B, =, AB, +, AB', =, AC, +, A'C, :=, AC, +, AC', =, BC, +, B'C, =, BC, +, BC', . The three splitters
concur In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liabilit ...
at the
Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurr ...
of the triangle. A
cleaver A cleaver is a large knife that varies in its shape but usually resembles a rectangular-bladed hatchet. It is largely used as a kitchen or butcher knife and is mostly intended for splitting up large pieces of soft bones and slashing through ...
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 In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
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 invoking 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 = rs. The area of a triangle can also be calculated from its semiperimeter and side lengths ''a, b, c'' using Heron's formula: :A = \sqrt. The circumradius ''R'' 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. The inradius is : r = \sqrt. The law of cotangents 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 ''a'' is :t_a= \frac. In a
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
, the radius of the excircle on the
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equ ...
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 ''a'' and ''b'' are the legs.


For quadrilaterals

The formula for the semiperimeter of a quadrilateral with side lengths ''a'', ''b'', ''c'' and ''d'' is :s = \frac. One of the triangle area formulas involving the semiperimeter also applies to tangential quadrilaterals, 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 In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version ( Bretschneider's formula) can be used with non-cycli ...
for the area of a
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
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 the following expression for the area of a general quadrilateral: : K = \sqrt ::= \sqrt . Here, , , , are the sides of the quadrilateral, is the semiperimeter, and and are any two opposite angles, sinc ...
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 polyto ...
quadrilaterals: : K = \sqrt , in which \alpha \, and \gamma \, 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 center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius.


Regular polygons

The area of a
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 polyto ...
regular polygon is the product of its semiperimeter and its apothem.


See also

* Semidiameter


References


External links

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