Irregular octagon with perimeter

In plane geometry , a perimeter is a circle that goes through all corner points of a polygon (polygon).

Such a perimeter does not exist for every polygon. In general, a convex polygon has a perimeter if and only if the perpendiculars of all sides intersect at a point. In this case the common point is the center of the perimeter.

## Perimeter of a triangle

Triangle with vertical lines and perimeter

The perimeter is particularly important in triangular geometry . Each triangle has a perimeter, as explained below.

All points of the perpendicular to are from and equidistant. Correspondingly, the points of the median perpendicular have to be equal distances from and . The point of intersection of these two perpendicular lines is therefore the same distance from all three corners ( , and ). It must therefore also be on the third vertical line. If you draw a circle around this intersection that goes through one corner of the triangle, the other corners must also lie on this circle. ${\ displaystyle [AB]}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle [BC]}$${\ displaystyle B}$${\ displaystyle C}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle C}$

The center of the circumference, i.e. the point of intersection of the vertical lines, is one of the excellent points of the triangle. He bears the Kimberling number . ${\ displaystyle X_ {3}}$

### special cases

For acute triangles , the center of the circumference is inside the triangle. In the case of a right triangle , the center of the hypotenuse is also the center of the circumference (see Thales theorem ). In the case of an obtuse triangle (with an angle greater than 90 °), the circumcenter is outside the triangle.

The perimeter radius of a triangle can be calculated using the sine law

${\ displaystyle R = {\ frac {a} {2 \ sin \ alpha}} = {\ frac {b} {2 \ sin \ beta}} = {\ frac {c} {2 \ sin \ gamma}}}$

or calculate from the triangular area:

${\ displaystyle R = {\ frac {abc} {4A}}}$.

Here are the names , , for the side lengths and , , for the sizes of the interior angles . denotes the area of the triangle, which z. B. can be calculated using the Heronic formula . ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle c}$${\ displaystyle \ alpha}$${\ displaystyle \ beta}$${\ displaystyle \ gamma}$${\ displaystyle A}$

### Coordinates

The Cartesian coordinates of the perimeter center can be calculated from the Cartesian coordinates of the corner points. The coordinates of the three corner points are , and . ${\ displaystyle (x_ {u}, y_ {u})}$${\ displaystyle (x_ {1}, y_ {1})}$${\ displaystyle (x_ {2}, y_ {2})}$${\ displaystyle (x_ {3}, y_ {3})}$

With

${\ displaystyle d = 2 (x_ {1} (y_ {2} -y_ {3}) + x_ {2} (y_ {3} -y_ {1}) + x_ {3} (y_ {1} -y_ {2}))}$

result in the Cartesian coordinates of the circumcenter

${\ displaystyle x_ {u} = {\ frac {(x_ {1} ^ {2} + y_ {1} ^ {2}) (y_ {2} -y_ {3}) + (x_ {2} ^ { 2} + y_ {2} ^ {2}) (y_ {3} -y_ {1}) + (x_ {3} ^ {2} + y_ {3} ^ {2}) (y_ {1} -y_ {2})} {d}}}$

and

${\ displaystyle y_ {u} = {\ frac {(x_ {1} ^ {2} + y_ {1} ^ {2}) (x_ {3} -x_ {2}) + (x_ {2} ^ { 2} + y_ {2} ^ {2}) (x_ {1} -x_ {3}) + (x_ {3} ^ {2} + y_ {3} ^ {2}) (x_ {2} -x_ {1})} {d}}}$.
Circumcenter of a triangle ( ) ${\ displaystyle X_ {3}}$
Trilinear coordinates ${\ displaystyle \ cos \ alpha \,: \, \ cos \ beta \,: \, \ cos \ gamma}$

${\ displaystyle = a (b ^ {2} + c ^ {2} -a ^ {2}) \,: \, b (c ^ {2} + a ^ {2} -b ^ {2}) \ ,: \, c (a ^ {2} + b ^ {2} -c ^ {2})}$

Barycentric coordinates ${\ displaystyle \ sin (2 \ alpha) \,: \, \ sin (2 \ beta) \,: \, \ sin (2 \ gamma)}$

### Other properties

• Like the center of gravity and the intersection of the height , the center of the circumference lies on the Euler straight line .
• According to the South Pole theorem, the center perpendicular of one side of the triangle intersects the bisector of the opposite angle always on the circumference.
• The distance between the center of the circumference and the center of the incircle is , where the circumradius and the incircle radius are ( Euler's theorem ).${\ displaystyle {\ sqrt {R (R-2r)}}}$${\ displaystyle R}$${\ displaystyle r}$
• The sum of the signed distances between the center of the circumference and the sides of the triangle is equal to the sum of the circumference and the incircle radius (see Carnot's theorem ).
• The principle of Trident represents a relationship between radius and inscribed ago

### Generalization: middle perpendicular theorem

The statement, the center perpendicular to the sides of the triangle that intersect at a point, in the synthetic geometry as a means solders set called. There it can be shown for more general affine planes, in which no concept of distance and therefore no "circles" are defined, that this sentence is equivalent to the theorem of height intersection . → See also the vertical intersection and the pre-Euclidean plane .

## Orbiting other polygons

While there is always a circumference with a triangle, this only applies in special cases to polygons with more than three corners.

Squares that have a radius are cyclic quadrilateral called. Special cases are isosceles trapezoids , including rectangles and squares .

Regardless of the number of corners, every regular polygon has a perimeter. The following applies to the perimeter radius of a regular corner with the length of the side : ${\ displaystyle n}$${\ displaystyle a}$

${\ displaystyle R = {\ frac {a} {2 \ sin {\ frac {180 ^ {\ circ}} {n}}}}}$

## Related terms

The inscribed circle of a polygon is a circle that touches all sides of that polygon. The inscribed circle of a triangle represents a particularly important special case. Together with the circumference and the three circles, it belongs to the special circles of triangular geometry.

If the definition of the circumference is transferred to (three-dimensional) space , one obtains the concept of the sphere , i.e. a sphere on which all corner points of a given polyhedron ( polyhedron ) lie.