How to find an angle formed by radii and arcs in a circle

Key takeaway: When a diagram mixes radii, central angles, and arcs, the key is to identify which angle is at the center and which ones sit on the circle. That tells you which circle rule applies.

• The angle shown is a central angle because its vertex is at the circle’s center.
• For a central angle, the angle measure equals the measure of its intercepted arc.
• So if the intercepted arc is $240^\circ$, then

$m\angle QRS = 240^\circ$

Final answer

$240^\circ$

---

Pitfalls the pros know 👇 A common mistake is to use the inscribed-angle rule and divide by 2. That only works when the vertex is on the circle, not at the center. Also check whether the diagram is asking for the major arc or the minor arc; that changes the number immediately.

What if the problem changes? If the same picture asked for an inscribed angle intercepting the same arc, the angle would be half of $240^\circ$, so it would be $120^\circ$. If the arc were the minor arc instead, you would first compute $360^\circ-240^\circ=120^\circ$ and then match the angle rule to the diagram.

Tags: central angle, intercepted arc, circle theorem