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: An angle with its vertex at the circle's center measures exactly the arc its two radii cut off.

Common stuck point: The procedure for central angle is the easy part; the trap is halving a central angle's arc. Asking "Is the angle's vertex exactly at the center of the circle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the angle's vertex exactly at the center of the circle?

Worked Examples

Example 1

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

Answer

The central angle is 130°130°.

First step

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.

Full solution

  1. 2
    The intercepted arc measures 130°130°. By the theorem, the central angle = intercepted arc = 130°130°.
  2. 3
    Verify the context makes sense: a central angle of 130°130° means the remaining arc on the other side is 360°−130°=230°360° - 130° = 230°, and the corresponding reflex central angle would be 230°230°. Both arcs and their central angles sum to 360°360° ✓.
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°3x°, 5x°5x°, and 4x°4x°. Find each arc and central angle.

Example 3

medium
A pie chart shows budget categories. Housing is 40%40\%, Food is 25%25\%, Transport is 15%15\%, Other is the rest. Find the central angle for Other.

Example 4

medium
A circular garden is divided into sectors of 90°90°, 120°120°, and the rest. What is the central angle of the remaining sector?

Example 5

hard
In circle OO with radius 1010, central angle ∠AOB=90°\angle AOB = 90°. Find the length of chord ABAB.

Example 6

hard
At 3:30 on an analog clock, what is the angle between the hour hand and the minute hand?

Example 7

challenge
In circle OO of radius 1010, three chords ABAB, BCBC, CACA form an equilateral triangle inscribed in the circle. Find the central angle subtended by each chord.

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 OO, central angle AOB=80°AOB = 80° and central angle BOC=110°BOC = 110°. Find arc ACAC going the short way (not through BB), and the reflex arc ACAC going through BB.

Example 3

easy
A central angle measures 45°45°. What is the measure of the arc it intercepts?

Example 4

easy
On an analog clock, the minute hand moves from 12 to 3. Through what central angle does it sweep?

Example 5

easy
Three central angles in a circle measure 90°90°, 130°130°, and xx. They make up the whole circle. Find xx.

Example 6

medium
In circle OO, central angles ∠AOB\angle AOB and ∠BOC\angle BOC measure 50°50° and 70°70°. Find ∠AOC\angle AOC, assuming BB is between AA and CC on the same arc.

Example 7

medium
Two central angles in a circle are in the ratio 2:32 : 3 and together intercept the entire circle. Find the larger angle.

Example 8

medium
In circle OO, ∠AOB=4x+20°\angle AOB = 4x + 20° and the arc ABAB measures 7x−10°7x - 10°. Find xx.

Example 9

medium
An inscribed angle measures 35°35°. What is the measure of the central angle that intercepts the same arc?

Example 10

medium
Convert a central angle of 60°60° into radians.

Example 11

medium
A sector has central angle 72°72°. What fraction of the circle does it represent?

Example 12

hard
In circle OO, chord ABAB subtends a central angle of 60°60°, and the radius is 88. Find the length of chord ABAB.

Example 13

hard
Four central angles in a circle measure xx, 2x2x, 3x3x, and 4x4x. Find the smallest angle.

Example 14

hard
Find the central angle in radians that subtends one-fifth of a circle.

Example 15

hard
In circle OO with radius 55, a chord has length 535\sqrt{3}. Find the central angle (acute) that subtends this chord.

Example 16

hard
Two central angles in a circle are supplementary (90°90° and 90°90° would be a degenerate case). If ∠AOB+∠COD=180°\angle AOB + \angle COD = 180° and ∠AOB=70°\angle AOB = 70°, find ∠COD\angle COD.

Example 17

challenge
A regular hexagon is inscribed in circle OO of radius rr. What is the central angle that one side of the hexagon subtends?
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Background Knowledge

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

circlesangles