Central angles, inscribed angles, and arcs
Central and Inscribed Angles
Central and Inscribed Angles is a topic in Circles in the Common Core State Standards. It is aligned to Standard G-C.2, which requires students to identify and describe relationships among inscribed angles, radii, and chords, including the inscribed angle being half the central angle.
A central angle equals its intercepted arc, and an inscribed angle is half of its intercepted arc.
Theory
Angles in a circle relate to the arcs they intercept:
- A central angle (vertex at the center) equals its intercepted arc.
- An inscribed angle (vertex on the circle) is half its intercepted arc.
- Inscribed angles that intercept the same arc are equal.
- An angle inscribed in a semicircle (subtending a diameter) is a right angle.
The angle-arc relationships:
How to find a circle angle
- Identify whether the angle is central or inscribed.
- Central: angle \(=\) arc.
- Inscribed: angle \(=\dfrac12\) arc.
- Special cases: same arc gives equal angles; a diameter gives \(90^\circ\).
A central angle equals its intercepted arc.
| \(\text{central angle}\) | \(=\) | \(80^\circ\) |
An inscribed angle is half its intercepted arc.
| \(\dfrac{80^\circ}{2}\) | \(=\) | \(40^\circ\) |
Inscribed angles on the same arc are equal.
| \(\text{other}\) | \(=\) | \(35^\circ\) |
An angle inscribed in a semicircle is a right angle.
| \(\text{angle}\) | \(=\) | \(90^\circ\) |
Common pitfalls
Frequently asked questions
What is the inscribed angle theorem?
An inscribed angle is half the central angle that subtends the same arc.
How does a central angle relate to its arc?
A central angle has the same measure as the arc it intercepts.
What is the measure of an angle inscribed in a semicircle?
\(90^\circ\) — an angle subtending a diameter is a right angle.
Are inscribed angles on the same arc equal?
Yes. Any inscribed angles intercepting the same arc have the same measure.