Detailed definition
Understanding Inscribed Angle
An inscribed angle has its vertex on the circle and its sides are chords that meet the boundary at that vertex. Because the vertex is on the circumference, the angle reads the circle from the edge rather than from the center.
The central relationship is that an inscribed angle measures half the measure of its intercepted arc in the standard minor-arc setting. This is also why an angle inscribed in a semicircle is always a right angle.
This page keeps the angle point and intercepted arc visible together so the theorem can be read from the geometry instead of memorised without context.
Key facts
Important ideas to remember
- An inscribed angle has its vertex on the circle and intercepts an arc.
- The sides of an inscribed angle are chords, not radii.
- An inscribed angle equals half the measure of its intercepted arc in the usual case.
- If the intercepted arc is a semicircle, the inscribed angle measures 90 degrees.
Where it is used
Where inscribed angle shows up
- Use inscribed angles in arc-measure and missing-angle problems.
- Use them when a circle theorem links an edge point on the circle to an arc across the figure.
- Use them when comparing the same endpoints under both a central angle and an inscribed angle.
Common mistakes
What to watch out for
- Do not confuse the inscribed-angle rule with the central-angle rule; the measures are not the same.
- Do not choose the wrong intercepted arc when tracing the angle's sides to the circle.
- Do not place the vertex inside the circle and still call the angle inscribed.