1. What is a 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 radius of the circle.

2. How Does the Calculator Work?

The calculator uses the central angle formula:

Where:

• \( \theta \) — Central angle in radians
• \( \text{arc\_length} \) — Length of the arc (same units as radius)
• \( \text{radius} \) — Radius of the circle

Explanation: The central angle in radians is equal to the ratio of the arc length to the radius of the circle.

3. Importance of Radians

Details: Radians are the standard unit of angular measure in mathematics. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.

4. Using the Calculator

Tips: Enter the arc length and radius in the same units (e.g., both in meters or both in inches). Both values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between radians and degrees?
A: Radians are based on the radius of a circle, while degrees divide a circle into 360 arbitrary units. 2π radians = 360 degrees.

Q2: Can I use this calculator for any circle?
A: Yes, as long as you know the arc length and radius, this works for any circle.

Q3: What if my arc length is greater than the circumference?
A: The calculator will still work, giving you an angle greater than 2π radians (360°), representing multiple rotations around the circle.

Q4: How do I convert radians to degrees?
A: Multiply radians by (180/π) to convert to degrees.

Q5: What are common applications of central angles?
A: Central angles are used in navigation, engineering, physics, and anywhere circular motion or geometry is involved.