Inscribed Angle in a Circle

Inscribed Angle in a Circle

An angle in a circle with vertex on the circle itself.

Circle with a point on its circumference forming an inscribed angle, with two chords extending to the circle's edge.

Key Formula

\theta = \frac{1}{2} \cdot \text{intercepted arc}

Where:

$\theta$ = The measure of the inscribed angle (in degrees)
$\text{intercepted arc}$ = The degree measure of the arc that lies in the interior of the angle, cut off by the angle's two sides

Worked Example

Problem: An inscribed angle in a circle intercepts an arc that measures 140°. What is the measure of the inscribed angle?

Step 1: Identify the intercepted arc. The arc cut off by the two sides of the inscribed angle measures 140°.

\text{intercepted arc} = 140°

Step 2: Apply the inscribed angle theorem: the inscribed angle equals half its intercepted arc.

\theta = \frac{1}{2} \cdot 140°

Step 3: Calculate the result.

\theta = 70°

Answer: The inscribed angle measures 70°.

Another Example

This example demonstrates two key corollaries: (1) inscribed angles intercepting the same arc are congruent, and (2) you must be careful to identify which arc (minor or major) the angle actually intercepts.

Problem: Two inscribed angles in the same circle both intercept arc PQ, which measures 110°. A third inscribed angle intercepts the remaining arc (the major arc PQ). Find all three angle measures.

Step 1: Both inscribed angles intercept the same arc PQ of 110°. By the inscribed angle theorem, inscribed angles that intercept the same arc are equal.

\theta_1 = \theta_2 = \frac{1}{2} \cdot 110° = 55°

Step 2: Find the major arc PQ. A full circle is 360°, so the remaining arc is:

\text{major arc } PQ = 360° - 110° = 250°

Step 3: The third inscribed angle intercepts this major arc, so apply the formula again.

\theta_3 = \frac{1}{2} \cdot 250° = 125°

Answer: The first two inscribed angles each measure 55°, and the third measures 125°.

Frequently Asked Questions

What is the difference between an inscribed angle and a central angle?

A central angle has its vertex at the center of the circle, while an inscribed angle has its vertex on the circle. For the same intercepted arc, a central angle equals the arc measure, but an inscribed angle equals half the arc measure. So a central angle is always twice the inscribed angle that intercepts the same arc.

Why is an inscribed angle half the central angle?

The inscribed angle theorem can be proven by drawing a radius to the vertex of the inscribed angle and using the fact that the base angles of an isosceles triangle (formed by two radii and a chord) are equal. Through this construction, the exterior angle of the triangle equals twice the inscribed angle, and that exterior angle equals the central angle. This relationship holds for all three possible cases (the center inside the angle, on one side, or outside).

What is the inscribed angle in a semicircle?

An inscribed angle that intercepts a semicircle (an arc of 180°) always measures exactly 90°. This is known as Thales' theorem. Any triangle inscribed in a circle where one side is a diameter will have a right angle opposite that diameter.

Inscribed Angle vs. Central Angle

	Inscribed Angle	Central Angle
Vertex location	On the circle	At the center of the circle
Formula	θ = ½ × intercepted arc	θ = intercepted arc
Relationship	Half the central angle for the same arc	Twice the inscribed angle for the same arc
Special case	90° when intercepting a semicircle	180° when intercepting a semicircle (a straight angle)

Why It Matters

Inscribed angles appear throughout geometry courses, especially in problems involving cyclic quadrilaterals, tangent lines, and circle theorems on standardized tests like the SAT and ACT. Understanding the inscribed angle theorem also provides the foundation for Thales' theorem (a right angle in a semicircle), which is used in constructions and proofs. Many real-world design problems—such as determining viewing angles in architecture or optics—rely on the properties of inscribed angles.

Common Mistakes

Mistake: Confusing the inscribed angle with the central angle and using the full arc measure as the angle.

Correction: The inscribed angle is always half the intercepted arc. Only a central angle equals the arc directly. If the arc is 120°, the inscribed angle is 60°, not 120°.

Mistake: Choosing the wrong arc when an inscribed angle is obtuse.

Correction: An inscribed angle intercepts the arc in its interior. If the vertex is on the major arc, the angle intercepts the minor arc (giving an acute angle). If the vertex is on the minor arc, the angle intercepts the major arc (giving an obtuse angle). Always trace the two chords from the vertex and identify which arc lies between their other endpoints, inside the angle.

Related Terms

Angle — General concept; inscribed angle is a specific type
Circle — The shape on which the inscribed angle is formed
Vertex — The point on the circle where the angle is formed
Central Angle — Angle at center; equals twice the inscribed angle
Arc — The intercepted arc determines the inscribed angle
Chord — The two sides of an inscribed angle are chords
Cyclic Quadrilateral — Uses inscribed angle properties; opposite angles sum to 180°