1. What is Central Angle?

A central angle is an angle whose apex (vertex) is the center of a circle and whose legs (sides) are radii intersecting the circle in two distinct points. The central angle is related to the arc length and circumference of the circle.

2. How Does the Calculator Work?

The calculator uses the central angle formula:

Where:

• \( \theta \) — Central angle in degrees
• \( \text{arc\_length} \) — Length of the arc (same units as circumference)
• \( \text{circumference} \) — Total circumference of the circle

Explanation: The formula calculates what portion of the full circle (360°) the arc length represents.

3. Importance of Central Angle Calculation

Details: Central angles are crucial in geometry, engineering, and design for calculating arc lengths, sector areas, and understanding circular motion.

4. Using the Calculator

Tips: Enter both arc length and circumference in the same units. Both values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What if I know the radius instead of circumference?
A: First calculate circumference using \( C = 2\pi r \), then use this calculator.

Q2: Can I calculate arc length from central angle?
A: Yes, rearrange the formula: \( \text{arc\_length} = (\theta / 360) \times \text{circumference} \).

Q3: What units should I use?
A: Any consistent units (cm, m, inches, etc.) as long as both measurements use the same units.

Q4: What's the range of possible central angles?
A: From 0° to 360°. Angles beyond 360° represent multiple full rotations.

Q5: How does this relate to radians?
A: To convert to radians: \( \theta_{\text{radians}} = (\theta_{\text{degrees}} \times \pi) / 180 \).