Central Angle Formula

The Formula

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

Notation

\angle AOB where O is the center; \overset{\frown}{AB} denotes the arc from A to B

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 \theta = m(\angle AOB) = m(\overset{\frown}{AB}) where O is the center; arc length s = r\theta (in radians); the central angle subtends a fraction \frac{\theta}{2\pi} of the circle

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.

Common Mistakes

  • Confusing central angle with inscribed angle (vertex at center vs on circle)
  • Measuring the wrong arc (major arc instead of minor arc)
  • Forgetting that the central angle and arc have the same degree measure

Why This Formula Matters

The starting point for all circle theorems. Central angles connect angles to arcs, which leads to arc length, sector area, and inscribed angle relationships.

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?

\angle AOB where O is the center; \overset{\frown}{AB} denotes the arc from A to B

Why is the Central Angle formula important in Math?

The starting point for all circle theorems. Central angles connect angles to arcs, which leads to arc length, sector area, and inscribed angle relationships.

What do students get wrong about Central Angle?

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

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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