Central angles, inscribed angles, and arcs
Central and Inscribed Angles
Central and Inscribed Angles is a topic in Circles in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.12(A), which requires students to apply theorems about the relationships between central angles, inscribed angles, and arcs.
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.