Inscribed Angle — Definition, Formula & Examples

An inscribed angle is an angle formed by two chords that meet at a point on the circle. Its measure is always half the measure of the arc it intercepts.

Given a circle with center $O$ and three points $A$ , $B$ , $C$ on the circle, the inscribed angle $\angle ABC$ is the angle at vertex $B$ formed by chords $\overline{BA}$ and $\overline{BC}$ . By the Inscribed Angle Theorem, $m\angle ABC = \tfrac{1}{2}\, m\overset{\frown}{AC}$ , where $\overset{\frown}{AC}$ is the intercepted arc not containing $B$ .

Key Formula

\theta = \frac{1}{2}\, m\overset{\frown}{AC}

Where:

$\theta$ = Measure of the inscribed angle
$m\overset{\frown}{AC}$ = Degree measure of the intercepted arc

How It Works

To find an inscribed angle, identify the vertex on the circle and the two sides (chords or secants) extending from it. The arc between the other two points, on the side of the circle opposite the vertex, is the intercepted arc. Divide that arc's degree measure by 2 to get the inscribed angle. A useful corollary: any inscribed angle that intercepts a semicircle (a 180° arc) is exactly 90°. Also, all inscribed angles that intercept the same arc are equal, regardless of where the vertex sits on the remaining arc.

Worked Example

Problem: Points A, B, and C lie on a circle. The intercepted arc AC (not containing B) measures 130°. Find the inscribed angle ABC.

Apply the Inscribed Angle Theorem: The inscribed angle equals half the intercepted arc.

m\angle ABC = \frac{1}{2} \times 130° = 65°

Answer: The inscribed angle ABC measures 65°.

Why It Matters

The Inscribed Angle Theorem appears repeatedly in high school geometry proofs and on standardized tests like the SAT. It is also essential in constructions and engineering problems that involve circular arcs, such as designing curved bridges or analyzing satellite dish geometry.

Common Mistakes

Mistake: Confusing an inscribed angle with a central angle and using the full arc measure instead of half.

Correction: A central angle equals its intercepted arc, but an inscribed angle is exactly half its intercepted arc. Always check whether the vertex is at the center or on the circle.