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Metric geometry : Angles on the circumference : Central and inscribed

angulo inscrito en una circunferenciaIn metric geometry measuring two concepts on which is based its axiomatic model: measures linear and angular measurements.
The linear measure abuts the Pythagorean theorem and relationship among these measures on Thales theorem.
The angular measurement from the relations expressed in a circle and with the above can describe the magnitude of geometric figures.
  • Central Angle -. It is that which has its apex at the center on the circumference and is understood as the arc.
  • Inscribed angle -. is one that has its vertex on the circle and its sides are chords.
angulos central e inscritos a circunferencia

Central and inscribed angles

An inscribed angle is half of the central angle covered by the same arc.

The sum of angles of a triangle is equal to two right, so the triangle CBP, that is isosceles (two equal angles) relationship will be fulfilled:


So we have that


And consequently deduce that the central angle is twice the enrollee


It is easy to generalize this concept to positions of point P that are not so private, and we can decompose the angle in two and apply the same reasoning.

Relacion angulos central e inscrito

Relationship between Central and inscribed angles

For example, if the point P move along the circumference, the central angle is the sum of the two central angles that can decompose, thus being indifferent point position P.


Central angle and inscribed



Examples inscribed angles

Metric geometry