Circumscribed circle

In plane geometry, a radius is a circle that passes through all the vertices of a polygon ( polygon ).

Not for each polygon exists such a radius. General has a convex polygon if and only within a radius when the perpendicular bisectors of all sides intersect at a point. In this case, the common point of the center of the circumference.

Perimeter of a triangle

Particularly great importance, the area in the triangle geometry. Each triangle has a perimeter, as justified below.

All points on the perpendicular bisector to are far from and the same. Accordingly have the points of the perpendicular bisectors to matching distances from and. The intersection of these two perpendicular bisectors is therefore by all three corners ( and ) the same distance. He must also be located on the third perpendicular bisector. If you draw to this intersection a circle that passes through a vertex of the triangle, as well as the other corners must lie on this circle.

The circumcenter, ie the intersection of the perpendicular bisectors, one of the excellent points of the triangle. He wears the Kimberling number.

Special cases

For acute-angled triangles the circumcenter lies inside the triangle. In the right-angled triangle the center of the hypotenuse circumcenter is at the same time (see theorem of Thales ). In the case of an obtuse triangle ( with an angle about 90 °) is the circumcenter outside the triangle.

Radius

The circumradius of a triangle can be combined with the Law of Sines

Or calculated from the triangle area.

Here are the names, and for the side lengths and, for the sizes of the interior angles. denotes the area of ​​the triangle, the example is with the help of the African hero formula can be calculated.

Coordinates

Other properties

• The circumcenter is like the center of gravity and the orthocenter to the Euler's line.
• According to the South Pole set is the perpendicular bisector of a side of the triangle intersects the bisector of the opposite angle is always on the perimeter.
• The distance between the circumcenter and incenter is where the circumradius and the Inkreisradius called ( Euler's theorem ).
• The sum of the signed distances from the circumcenter point of the triangle sides is equal to the sum of the radius and Inkreisradius (see Carnot's theorem ).

Generalization: Medium solders set

The statement that intersect the perpendicular bisectors of the sides of the triangle in one point is referred to in the synthetic geometry as a means sounding record. There, for more general affine planes in which no notion of distance, and thus no "circles" are defined, be shown that this set is equivalent to the height of the intersection set. → See orthocenter and präeuklidische level.

Perimeters of other polygons

During the triangle always exists a radius, this is true for polygons ( polygons ) with more than three vertices only in special cases.

Rectangles that have a perimeter are called tendons quadrangles. Special cases are isosceles trapezoids, including rectangles and squares.

Regardless of the number of vertices of each polygon has a regular radius. For the circumradius of a regular pentagon with side length applies:

Related terms

The inscribed circle of a polygon is a circle which touches all sides of this polygon. The inscribed circle of a triangle is a particularly important special case dar. He belongs to the perimeter and the three Ankreisen to the particular sections of the triangle geometry.

Applying the definition of the perimeter of the ( three-dimensional ) space, we obtain the notion of circumscribed sphere, ie a sphere on which all vertices of a given polyhedron ( polyhedron ) are.

Linguistic means " within "

In a geographical sense, is often spoken of a radius. Example: " The area within a radius of 10 km around the Chernobyl reactor was evacuated. " The so -intentioned circle has a radius of 10 kilometers and a diameter of 20 kilometers. He referred to the geographical area.