Degree to Radian Converter

Converting between degrees and radians is essential for trigonometry, calculus, and physics. This converter provides instant, precise results for any angle measurement. Whether you’re working on homework, engineering projects, or scientific research, get your conversions sorted in seconds.

Enter Angle in Degrees (°)

Decimal Places

Quick Conversions

Degrees to Radians Conversion Table

Here are the most commonly used angle conversions. These values appear frequently in trigonometry, geometry, and physics calculations.

Degrees (°)	Radians (exact)	Radians (decimal)
0°	0	0
30°	π/6	0.5236
45°	π/4	0.7854
60°	π/3	1.0472
90°	π/2	1.5708
120°	2π/3	2.0944
135°	3π/4	2.3562
150°	5π/6	2.6180
180°	π	3.1416
270°	3π/2	4.7124
360°	2π	6.2832

Conversion Formula

To convert any angle from degrees to radians, multiply by π/180:

Radians = Degrees × (π ÷ 180)

This works because a complete circle contains 360° or 2π radians. Therefore, 180° equals π radians, giving us the conversion ratio.

Step-by-Step Conversion Method

Take your angle measurement in degrees
Multiply the value by π (approximately 3.14159)
Divide the result by 180
The answer is your angle in radians

Example: Convert 45° to radians

45 × π ÷ 180 = 45π ÷ 180 = π/4 ≈ 0.7854 radians

Visual Comparison

A full circle contains 360 degrees or 2π radians. Here’s how they relate:

One radian is the angle created when the arc length equals the radius. This makes radians the natural choice for mathematical work.

When to Use Radians vs Degrees

Degrees suit everyday situations and are easier to visualise. Most people naturally think in degrees when describing angles. However, radians become essential in advanced maths and physics.

Use degrees for: Navigation, carpentry, architecture, general angle descriptions, and introductory geometry.

Use radians for: Calculus, trigonometric functions, physics equations, engineering calculations, and computer graphics programming.

Radians make formulae cleaner. The derivative of sin(x) is cos(x) only when x is in radians. Arc length equals radius times angle only with radians. Many scientific calculators default to radian mode for these reasons.

Where You’ll Need This

Radian conversions appear throughout science and maths. Physics uses radians for angular velocity, rotational motion, and wave equations. Engineering fields apply them in signal processing, control systems, and structural analysis.

In computer graphics and game development, rotation calculations rely on radians. Animation software often requires radian inputs for smooth motion. Robotics and navigation systems process angles in radians for precision.

Students encounter degree-to-radian conversions in A-level maths, particularly in trigonometry and calculus modules. University courses in physics, engineering, and computer science use radians as standard. Scientific research papers typically present angular measurements in radians.

Angle Conversion Reference

Besides degrees and radians, other angle measurements exist. Here’s how they all connect:

Unit	Symbol	Full Circle	Notes
Degree	°	360°	Most common in everyday use
Radian	rad	2π ≈ 6.2832	Standard for mathematical work
Gradian	ᵍ	400ᵍ	Used in surveying
Turn	tr	1 tr	One complete rotation

Worked Examples

Example 1: Convert 120° to radians

120 × π ÷ 180 = 120π ÷ 180 = 2π/3 ≈ 2.0944 radians

Example 2: Convert 225° to radians

225 × π ÷ 180 = 225π ÷ 180 = 5π/4 ≈ 3.9270 radians

Example 3: Convert 330° to radians

330 × π ÷ 180 = 330π ÷ 180 = 11π/6 ≈ 5.7596 radians

FAQs

Why do we use radians instead of degrees in calculus?

Radians make calculus formulae work properly. The derivative of sin(x) equals cos(x) only when x is measured in radians. Degrees would require extra conversion factors in every equation, making maths unnecessarily complicated. Radians are the natural unit for angles in mathematical analysis.

How many radians are in a circle?

A complete circle contains 2π radians, which equals approximately 6.2832 radians. This comes from the circumference formula: a circle’s circumference is 2πr, so when the radius sweeps the full circumference, it travels 2π radians.

Can you have negative radians?

Yes, negative radians indicate rotation in the opposite direction. Positive angles rotate anticlockwise (the standard mathematical direction), whilst negative angles rotate clockwise. The conversion formula works the same way for negative values.

What’s the quickest way to convert degrees to radians?

Multiply your degree value by 0.0174533 (which is π/180). This gives you radians directly. For common angles, it’s faster to memorise that 90° = π/2, 180° = π, and 360° = 2π, then work from there.

Do radians have units?

Technically, radians are dimensionless. They represent a ratio between arc length and radius, so the units cancel out. However, we still write “rad” to clarify we’re measuring angles, not just stating a number.

How do I convert radians back to degrees?

Multiply the radian value by 180/π. This reverses the degree-to-radian conversion. For example, π/4 radians × 180/π = 45°. Many scientific calculators have a button for this conversion.

Why is π involved in the conversion?

Because π is the ratio between a circle’s circumference and diameter. Since radians measure angles based on arc length, and circles involve π in their geometry, π naturally appears in the conversion between the two systems.

Are there angles larger than 2π radians?

Absolutely. Angles can be any size. Values larger than 2π represent multiple complete rotations. For instance, 4π radians means two full circles, whilst 10 radians is about 1.6 rotations. This matters in physics when tracking objects that spin multiple times.