Central Angle Formula

Central angle is an angle whose vertex is at the center of a circle, with its two rays intersecting the circle at two points.

The Formula

Central angle=intercepted arc (in degrees)\text{Central angle} = \text{intercepted arc (in degrees)}

When to use: 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.

Quick Example

A central angle of 90°90° intercepts an arc that is 90360=14\frac{90}{360} = \frac{1}{4} of the circle: Central angle=intercepted arc=90°\text{Central angle} = \text{intercepted arc} = 90°

Notation

∠AOB\angle AOB where OO is the center; AB⌢\overset{\frown}{AB} denotes the arc from AA to BB

What This Formula Means

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.

Formal View

Central angle θ=m(∠AOB)=m(AB⌢)\theta = m(\angle AOB) = m(\overset{\frown}{AB}) where OO is the center; arc length s=rθs = r\theta (in radians); the central angle subtends a fraction θ2π\frac{\theta}{2\pi} 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.

Common Mistakes

  • Halving a central angle's arc — only inscribed angles take half; a central angle equals its full arc.
  • Confusing the central angle (degrees) with the arc length (distance) — same arc, different kind of measurement.
  • Placing the vertex on the circle and still calling it central — central means the vertex is at the center.

Why This Formula Matters

It is the baseline for all circle-angle reasoning: arc length, sector area, and the inscribed-angle theorem are all defined relative to the central angle, and the whole chain of circle theorems collapses if you mistake a center vertex for an on-circle vertex. Recognizing it by "Is the angle's vertex exactly at the center of the circle?" — rather than by familiar numbers — is what lets a student tell it apart from inscribed angle and arc length and sector area in a mixed problem set.

Frequently Asked Questions

What is the Central Angle formula?

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.

How do you use the Central Angle formula?

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.

What do the symbols mean in the Central Angle formula?

∠AOB\angle AOB where OO is the center; AB⌢\overset{\frown}{AB} denotes the arc from AA to BB

Why is the Central Angle formula important in Math?

It is the baseline for all circle-angle reasoning: arc length, sector area, and the inscribed-angle theorem are all defined relative to the central angle, and the whole chain of circle theorems collapses if you mistake a center vertex for an on-circle vertex. Recognizing it by "Is the angle's vertex exactly at the center of the circle?" — rather than by familiar numbers — is what lets a student tell it apart from inscribed angle and arc length and sector area in a mixed problem set.

What do students get wrong about Central Angle?

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.

What should I learn before the Central Angle formula?

Before studying the Central Angle formula, you should understand: circles, angles.

Want the Full Guide?

This formula is covered in depth in our complete guide:

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