Inscribed Angle Calculator

Calculate inscribed angles, central angles, and intercepted arcs using the inscribed angle theorem.

Select What to Find

Result

Inscribed Angle
60
degrees
Inscribed Angle60
Central Angle120
Intercepted Arc120
Remaining Arc240
Inscribed (radians)1.0472

Step-by-Step Solution

Inscribed = Central / 2 = 120 / 2 = 60 degrees

The Inscribed Angle Theorem

The inscribed angle theorem states that an inscribed angle is half the central angle that subtends the same arc. Equivalently, the inscribed angle is half the measure of the intercepted arc. This is one of the most important theorems in circle geometry.

An inscribed angle is an angle formed by two chords that share an endpoint on the circle. The vertex of the inscribed angle lies on the circle, and the two sides of the angle are chords of the circle. The intercepted arc is the arc that lies in the interior of the inscribed angle.

Key Relationships

Inscribed Angle Theorem

The inscribed angle is exactly half the central angle subtending the same arc.

inscribed = central / 2

Central Angle from Inscribed

The central angle is twice the inscribed angle.

central = 2 x inscribed

Arc and Central Angle

The central angle equals the measure of the intercepted arc.

arc = central angle

Semicircle Rule

An inscribed angle that intercepts a semicircle (180 arc) is always 90 degrees.

inscribed = 180 / 2 = 90

Special Cases and Corollaries

  • Thales' Theorem: An inscribed angle in a semicircle is always a right angle (90 degrees).
  • Equal Inscribed Angles: Inscribed angles that intercept the same arc are equal, regardless of where the vertex is on the circle.
  • Opposite Angles in Cyclic Quadrilateral: In a quadrilateral inscribed in a circle, opposite angles sum to 180 degrees.
  • Tangent-Chord Angle: The angle between a tangent and a chord equals half the intercepted arc (similar to inscribed angle theorem).

Practical Applications

The inscribed angle theorem is used in navigation (finding positions using circular arcs), architecture (designing arched structures), astronomy (calculating apparent angles), and computer graphics (rendering circular shapes). It is also fundamental in the proof of many other geometric theorems.

Tips for Problem Solving

  • Always identify the intercepted arc first when working with inscribed angles.
  • Remember that the central angle and its intercepted arc always have the same measure in degrees.
  • If you know the inscribed angle, multiply by 2 to get the central angle or arc.
  • An inscribed angle can never exceed 180 degrees; a central angle can be up to 360 degrees.