Circles

Central & Inscribed Angles

Relating central angles, inscribed angles, and their intercepted arcs.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Two Ways to Measure the Same Arc

An angle with its vertex at a circle's center intercepts a particular arc. A different angle, with its vertex on the circle itself, intercepts that exact same arc. Before reading on, predict: do you think these two angles are equal, or is there some fixed relationship — like one being a specific multiple of the other?

Definition — Central Angle, Inscribed Angle, and Arc Measure

A central angle has its vertex at the circle's center; its measure equals the measure of the arc it intercepts. An inscribed angle has its vertex on the circle itself, with both sides as chords. The Inscribed Angle Theorem: an inscribed angle's measure is always exactly half the measure of its intercepted arc — equivalently, half the corresponding central angle.

Here's why, at least in the simplest case, where one side of the inscribed angle happens to pass through the center. Draw a radius from the center to the inscribed angle's vertex, forming an isosceles triangle (two sides are radii, hence equal). The central angle intercepting the same arc is an exterior angle of this triangle, so it equals the sum of the two non-adjacent interior angles — and since the triangle is isosceles, those two interior angles are equal to each other, and one of them is exactly the inscribed angle. That forces the central angle to equal exactly twice the inscribed angle, which is the theorem.

Worked Example — Finding an Arc from a Central Angle

A central angle measures 84°. Find the measure of its intercepted arc. A central angle's measure always equals its intercepted arc's measure directly: the arc measures 84°.

Worked Example — Finding an Inscribed Angle from an Arc

An inscribed angle intercepts an arc measuring 100°. Find the inscribed angle. By the Inscribed Angle Theorem, the inscribed angle is half the arc: 100° ÷ 2 = 50°.

Worked Example — An Inscribed Angle in a Semicircle

An inscribed angle intercepts a semicircle (an arc of exactly 180°, since it's half the full circle). Find the inscribed angle. By the theorem: 180° ÷ 2 = 90°. Every inscribed angle that intercepts a semicircle is a right angle — a useful corollary worth remembering on its own.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

Whenever a triangle is inscribed in a circle with one side as a diameter, that triangle automatically contains a right angle, opposite the diameter — a direct, frequently useful consequence of the Inscribed Angle Theorem.

Common Mistakes

Treating an inscribed angle's measure as equal to its intercepted arc, the same rule that applies to central angles.
Only central angles equal their intercepted arc directly — an inscribed angle is always exactly half of its intercepted arc's measure, not equal to it.
Misidentifying which arc an inscribed or central angle actually intercepts, especially when a circle has multiple angles drawn on it.
Trace each side of the angle out to where it meets the circle — the intercepted arc is specifically the arc 'cut off' between those two points, on the side away from the angle's interior.

Key Takeaways

A central angle equals its intercepted arc's measure directly.
An inscribed angle equals exactly half its intercepted arc's measure, provable using an isosceles triangle and the exterior angle relationship.
Every inscribed angle intercepting a semicircle is a right angle.

Summary

Central and inscribed angles connect angle measure to arc measure through a provable, fixed relationship. The next lesson extends circle reasoning to lengths formed by chords, tangent lines, and secant lines.

Track your progress

Back to Circles Chords, Tangents & Secants→