Inscribed circle

The inscribed circle of a polygon ( polygon ) is the circle that touches all sides of the polygon in its interior ( ie it touches the distances between the vertices and not their extensions ). It is also the largest circle that lies entirely in the given polygon.

Only those polygons which intersect all bisectors of the interior angles of the polygon in a point, have an inscribed circle. The intersection in this case is the center of the inscribed circle.

If there is the inscribed circle of a polygon with area and perimeter, the Inkreisradius has the value

Incircle of a triangle

Particularly great importance has the inscribed circle in the triangle geometry. Each triangle has an inscribed circle, its center located at the intersection of the three angle bisectors. If you draw to this intersection a circle tangent to one side of the triangle, so this circle touches the other two sides.

All points on the bisector of the interior angle at the same distance from the sides and. Corresponding to the angle bisector of the points have the same distance from and. The intersection of the two bisectors is therefore from all three sides of the triangle ( and ) the same distance. He must also be located on the third bisector.

The inscribed circle touches all three sides in the interior - in contrast to the three Ankreisen, each touching the outside of one side and the extensions of the other two sides.

The incenter, so the intersection of the bisector, is one of the excellent points of the triangle. He wears the Kimberling number.


If the area of ​​the triangle with the sides, and so the radius of the inscribed circle is calculated by:


Depending on the given parameters of the triangle, the following relationship is interesting:


The Cartesian coordinates of the inscribed circle center point is calculated as the weighted with the side lengths of the opposite sides of the middle vertex coordinates. If the three vertices in, and are and who the vertices opposite sides of the lengths, and then there is the inscribed circle center point at


Other properties

  • The distance between the corner A and the adjacent contact points of the inscribed circle is the same; here means above half the circumference. The same applies to the vertices B and C.
  • The line connecting the corners of the triangle to the opposite points of contact of the inscribed circle intersect at a point, the Gergonne point.

Incircles other polygons

While triangles always exists an inscribed circle, this is true for polygons ( polygons ) with more than three vertices only in special cases.

Quadrilaterals have an inscribed circle, hot tangent squares. They include all convex quadrilaterals dragon, especially all diamonds and squares.

Regular polygons have - regardless of the number of corners - always an inscribed circle. For the Inkreisradius of a regular pentagon with side length applies: