Unit Circle Angle Converter
Instructions for Use:
- Enter an Angle value (e.g., 45, 3.14).
- Select the unit (Degrees or Radians).
- Click the “Convert” button.
- The converted angle will be displayed below, in the other unit.
The Unit Circle Angle Converter is a tool that helps you convert angles between degrees and radians, which are two commonly used units for measuring angles on the unit circle, particularly in trigonometry.
What is the Unit Circle?
The unit circle is a circle with a radius of 1, centered at the origin (0, 0) of the coordinate plane. The angle formed by a radius (a line from the center to the edge of the circle) and the positive x-axis is called the central angle. This angle can be measured in two common units: degrees and radians.
- Degrees: A full circle is divided into 360 degrees, meaning 360° is one complete revolution around the circle.
- Radians: A full circle is divided into 2π radians, which also represents one full revolution around the unit circle.
How to Convert Between Degrees and Radians
1. Degrees to Radians
To convert an angle from degrees to radians, use this formula:
- Radians = Degrees × (π / 180)
Explanation: Since 180° is equal to π radians, you multiply the angle in degrees by the fraction (π/180).
2. Radians to Degrees
To convert an angle from radians to degrees, use this formula:
- Degrees = Radians × (180 / π)
Explanation: Since π radians is equal to 180°, you multiply the angle in radians by the fraction (180/π).
Examples of Conversion
1. Degrees to Radians
- Convert 45° to radians: 45° × (π / 180) = π / 4So, 45° is equal to π/4 radians.
- Convert 120° to radians: 120° × (π / 180) = 2π / 3So, 120° is equal to 2π/3 radians.
2. Radians to Degrees
- Convert π/3 radians to degrees: (π / 3) × (180 / π) = 60°So, π/3 radians is equal to 60°.
- Convert π/4 radians to degrees: (π / 4) × (180 / π) = 45°So, π/4 radians is equal to 45°.
Unit Circle and Angle Conversions
The unit circle helps visualize angles, and here are some common angle conversions between degrees and radians:
Key Angles in Degrees and Radians
| Angle (Degrees) | Angle (Radians) | Coordinates on the Unit Circle |
|---|---|---|
| 0° | 0 | (1, 0) |
| 30° | π/6 | (√3/2, 1/2) |
| 45° | π/4 | (√2/2, √2/2) |
| 60° | π/3 | (1/2, √3/2) |
| 90° | π/2 | (0, 1) |
| 120° | 2π/3 | (-1/2, √3/2) |
| 180° | π | (-1, 0) |
| 270° | 3π/2 | (0, -1) |
| 360° | 2π | (1, 0) |
Why Use the Unit Circle Angle Converter?
- Trigonometry:
Trigonometric functions like sine, cosine, and tangent often involve degrees or radians. Converting between these units simplifies solving problems involving these functions. - Geometry and Physics:
Many formulas in geometry and physics use radians. Converting between degrees and radians ensures compatibility and accuracy in calculations. - Programming:
Many programming languages (like JavaScript, Python, and others) use radians for trigonometric functions. Understanding how to convert between angles is essential when coding for graphical applications, simulations, or even simple math operations.