Convert The Angle in Degrees to Radians Calculator

Converting angles from degrees to radians is a fundamental skill in mathematics, physics, and engineering. This calculator provides an easy way to perform the conversion while explaining the underlying formula and its practical applications.

How to Convert Degrees to Radians

The conversion from degrees to radians is straightforward once you understand the relationship between these two units of angular measurement. Degrees are based on the full circle being 360°, while radians are based on the full circle being 2π radians.

To convert degrees to radians, you multiply the degree measurement by π (pi) and then divide by 180. This formula comes from the fact that a full circle (360°) is equivalent to 2π radians.

Conversion Formula

Radians = Degrees × (π / 180)

For example, to convert 90° to radians:

Example Conversion

90° × (π / 180) = π/2 radians ≈ 1.5708 radians

Why Convert Degrees to Radians

While degrees are commonly used in everyday contexts, radians are the standard unit of angular measurement in many scientific and mathematical calculations. Radians are particularly useful in calculus, physics, and engineering because they simplify many formulas and equations.

For example, in trigonometric functions like sine and cosine, radians are used as the standard input. Many mathematical identities and formulas are more elegant and easier to work with when using radians.

Key Difference

Degrees are based on the full circle being 360°, while radians are based on the full circle being 2π radians. This means that 1 radian is approximately 57.2958°.

Common Angles in Degrees and Radians

Here are some common angles expressed in both degrees and radians:

Degrees	Radians	Approximate Value
0°	0	0
30°	π/6	0.5236
45°	π/4	0.7854
60°	π/3	1.0472
90°	π/2	1.5708
180°	π	3.1416
270°	3π/2	4.7124
360°	2π	6.2832

Conversion Formula

The formula for converting degrees to radians is:

Radians = Degrees × (π / 180)

Where π (pi) is approximately 3.141592653589793.

This formula works because a full circle is 360° and also 2π radians. Therefore, to convert from degrees to radians, you multiply by π/180.

Worked Example

Let's convert 120° to radians using the formula:

Step-by-Step Conversion

1. Start with the degree measurement: 120°

2. Multiply by π/180: 120 × (π / 180)

3. Simplify the fraction: 120/180 × π = 2/3 × π

4. Final result: 2π/3 radians ≈ 2.0944 radians

This means that 120° is equivalent to approximately 2.0944 radians.

Frequently Asked Questions

Why do we need to convert degrees to radians?

Radians are the standard unit of angular measurement in many scientific and mathematical calculations. They simplify many formulas and equations, particularly in calculus, physics, and engineering.

How do I convert radians back to degrees?

To convert radians back to degrees, use the formula: Degrees = Radians × (180 / π). This is the inverse of the degrees to radians conversion formula.

What is the difference between degrees and radians?

Degrees are based on the full circle being 360°, while radians are based on the full circle being 2π radians. This means that 1 radian is approximately 57.2958°.

When should I use degrees and when should I use radians?

Degrees are commonly used in everyday contexts, such as measuring angles in geometry or navigation. Radians are used in more advanced mathematics, physics, and engineering, where they simplify calculations and formulas.