Central Angle Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Central Angle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An angle whose vertex is at the center of a circle, with its two rays intersecting the circle at two points. Its measure equals the measure of the intercepted arc.

Imagine standing at the center of a clock face. The angle between the hour and minute hands is a central angle. The arc between the two numbers the hands point to is the intercepted arc, and its measure (in degrees) equals the angle you see.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A central angle equals its intercepted arc—they are the same measurement, just viewed differently.

Common stuck point: The central angle equals the arc it 'cuts off.' This is the baseline that other angle-arc relationships compare to.

Sense of Study hint: When you see a central angle problem, first confirm the vertex is at the center of the circle. Then set the angle measure equal to the intercepted arc measure. Finally, use s = r\theta if you need the actual arc length.

Worked Examples

Example 1

easy
A central angle in a circle intercepts an arc of 130°. What is the measure of the central angle?

Solution

  1. 1
    The Central Angle Theorem states that a central angle (vertex at the centre) is equal in measure to its intercepted arc. This is because both the angle and the arc are defined by the same two radii.
  2. 2
    The intercepted arc measures 130°. By the theorem, the central angle = intercepted arc = 130°.
  3. 3
    Verify the context makes sense: a central angle of 130° means the remaining arc on the other side is 360° - 130° = 230°, and the corresponding reflex central angle would be 230°. Both arcs and their central angles sum to 360° ✓.

Answer

The central angle is 130°.
A central angle is an angle whose vertex is at the center of the circle and whose sides are radii. By definition, the central angle and its intercepted arc have the same degree measure. This is the foundational theorem for circle arc-angle relationships.

Example 2

medium
In a circle, three central angles divide the circle into three arcs with measures 3x°, 5x°, and 4x°. Find each arc and central angle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A circle is divided into 6 equal sectors. What is the central angle of each sector?

Example 2

hard
In circle O, central angle AOB = 80° and central angle BOC = 110°. Find arc AC going the short way (not through B), and the reflex arc AC going through B.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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