Semiperimeter

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Semiperimeter

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the semiperimeter of a polygon is half its perimeter

Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
. 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

The area of any triangle is the product of its inradius and its semiperimeter; the same area formula also applies to tangential quadrilateral

Tangential quadrilateral

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Encyclopedia

In geometry

Geometry

Perimeter

Tangential quadrilateral

In geometry, a tangential quadrilateral is a convex polygon quadrilateral whose sides all lie tangent to a single circle inscribed within the quadrilateral....
s, 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

Heron's formula

In geometry, Heron's formula states that the area of a triangle whose sides have lengths a, b, and c iswhere s is the semiperimeter of the triangle:...
:

The simplest form of Brahmagupta's formula

Brahmagupta's formula

In geometry, Brahmagupta's formula finds the area of any quadrilateral given the lengths of the sides and some of their angles. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle....
, for the area of a cyclic quadrilateral

Cyclic quadrilateral

In geometry, a cyclic quadrilateral is a quadrilateral whose vertex all lie on a single circle. The vertices are said to be concyclic.In a cyclic quadrilateral, opposite angles are supplementary angle ....
, has a similar form:

The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths: This formula can be derived from the law of sines

Law of sines

The law of sines , in trigonometry, is a statement about any triangle in a plane. Where the sides of the triangle are a, b and c and the angles opposite those sides are A, B and C, then the law of sines states equality of the first three quantities below:...
.

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

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

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

Nagel point

In geometry, the Nagel point is a point associated with any triangle. In the plane of a triangle ABC with side lengths a = |BC|, b = |CA|, and c = |AB|, let TA, TB, and TC be the points in which the A-excircle meets line BC, the B-excircle meets...
of the triangle.

External links

by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".