# Semiperimeter

In geometry , the perimeter of a polygon is half its perimeter . Although it has such a simple derivation from the perimeter, the semiperimeter appears often enough in formulas for triangles and other figures that it is given a different name. When the semiperimeter occurs as part of a formula, it is usually represented by the letter s .

## Triangle

Semiperimeter is often used for triangles; Formula for the perimeter of a triangle whose sides are of length a , b , and c

s={\frac {a+b+c}{2}}.


### Virtue

In any triangle, any vertex and the point where the triangle touches the outflow in the opposite triangle divides the perimeter of the triangle into two equal lengths, thus creating two paths, each of which has length equal to the semi-perimeter. If a, b, c, a’, b’, and c’ as shown in the figure, then the segments connecting a vertex with opposite circle tangents (AA’, BB’, and CC’, in red) shown 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 agree on the Nagel point of the triangle.

A cleaver is a line segment of a triangle that bisects the perimeter of the triangle and has an end point in the middle of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths, each of length equal to the semiperimeter. The three cleavers agree on the center of the SPIEKER circle , which is the incircle of the medial triangle ; The spiker center is the center of mass of all points on the sides of the triangle .

A line through the center of a triangle bisects the perimeter if and only if it also bisects the area. The perimeter of a triangle is equal to the circumference of its medial triangle.

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

### Formulas invoking semiparameters

The area a of any triangle is the product of its inradius (the radius of its inscribed circle) and its semi-perimeter:

A = rs.

The area of ​​a triangle can also be calculated from its semiperimeter and the lengths of the sides a , b, c using Heron’s formula :

A = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.


The perimeter R of a triangle can also be calculated from the semiperimeter and the length of the side:

R={\frac {abc}{4{\sqrt {s(s-a)(s-b)(s-c)}}}}.


This formula can be obtained from the law of sine .

r={\sqrt {{\frac {(s-a)(s-b)(s-c)}{s}}}}.

The arrangement of cotangents gives the cotangents of half the angles in the vertices of a triangle in terms of the semiperimeter, sides, and inradius.

The length of the internal bisector of the angle opposite to the side of length a is

t_{a}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}.


In a right angled triangle , the radius of the circle on the hypotenuse is equal to the semi-perimeter . The semiperimeter is the sum of the inradius and twice the circumference. The area of ​​the right angled triangle is (s-a)(s-b) where a and b are legs.

The formula for the perimeter of a quadrilateral with side lengths a , b , c and d is

s={\frac {a+b+c+d}{2}}.

One of the formulas for the triangle area involving semiperimeters also applies to tangent quadrilaterals , which have an incircle and in which ( according to Pitot’s theorem ) the lengths of pairs of opposite sides are the sum of the semiperimeters—that is, the area incircle. is the product of and semiparameter:

K=rs.


The simplest form of Brahmagupta’s formula for the area of ​​a cyclic quadrilateral is the same as Heron’s formula for the triangle area:

K={\sqrt {\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)}}.


Bretschneider’s formula generalizes this to all convex quadrilaterals:

K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}},

in which and are two opposite angles.

\alpha \,\gamma \,

The four sides of a bicentral quadrilateral parametrized by the semiperimeter, the interradius and the circumradius are the four solutions of the quartile equation .

### Regular polygon

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

#### What is the formula for the circumference of a half bridge?

Step-by-step explanation: Remember, the circumference of a circle can be found using the formula c = 2πr, where r is the radius of the circle.

#### What is the formula for semi perimeter?

The perimeter of a triangle is found by the formula P=a+b+c, where a, b and c are the sides of the triangle.

#### How to find circumference?

The formula for finding the perimeter is:
C = D; But what do they mean? This formula has three elements: C: C means circumference. You want to calculate the value of this; D: D means diameter of the circle. ,
The length of the circumference of a circle is pi times the diameter of that circle.

#### What is the angle of the semicircle?

Every angle of a semi circle is a right angle; That is, it is 90 degrees.

#### What is the radius of the semicircle?

If you are given the diameter of the semicircle, you can divide it by two to get the radius. For example, if the length of the diameter of a semicircle is 10 cm, dividing it by 2 (10/2) will make its radius 5 cm.

#### What is the measure of fasting?

Roughly speaking, the outer circle of the circle and the ellipse, and the length of the circle, is called the circumference. But generalizing it, the total length (perimeter) of the edges of any closed curve is called ‘perimeter’. That is, circumference is a particular state of perimeter.

#### What will be the perimeter of the circle?

circumference of circle
The distance around the circle is called the circumference or circumference of the circle. Pi (π ): This is a number equal to 3.141592… or 22/7. Pie (π) = (Perimeter) / (Diameter) of a circle.

#### How many degrees is the angle subtended on the circle?

The angle subtended by the diameter of a circle at any point on the circumference is a right angle (90 degrees). The perpendicular placed on a chord from the center is also the bisector of that chord.