Convert Radians and Degrees Calculator

Convert between radians and degrees with our precise calculator. Learn the conversion formula, see worked examples, and understand when to use each unit.

Conversion Formula

The relationship between radians and degrees is defined by the formula:

Degrees = Radians × (180/π)

Radians = Degrees × (π/180)

Where π (pi) is approximately 3.141592653589793. This conversion is based on the fact that a full circle is 360 degrees or 2π radians.

When to Use Radians

Radians are commonly used in:

Calculus and higher mathematics
Physics (especially rotational motion)
Engineering (especially electrical engineering)
Computer graphics and game development

Radians are the natural unit for circular functions like sine and cosine because they directly relate to the radius of a circle.

When to Use Degrees

Degrees are commonly used in:

Everyday measurements (like temperature)
Navigation and geography
Surveying and construction
Basic trigonometry problems

Degrees are more intuitive for humans because they divide a circle into 360 equal parts, making them easier to visualize.

Worked Example

Let's convert 45 degrees to radians:

Radians = 45 × (π/180)

Radians ≈ 45 × 0.0174533

Radians ≈ 0.785398 radians

This means 45 degrees is approximately 0.785 radians. You can verify this using our calculator by entering 45 in the degrees field and clicking "Calculate".

FAQ

How do I convert radians to degrees?: Multiply the radian value by 180/π. For example, 1 radian × 180/π ≈ 57.2958 degrees.
How do I convert degrees to radians?: Multiply the degree value by π/180. For example, 90 degrees × π/180 ≈ 1.5708 radians.
When should I use radians instead of degrees?: Use radians when working with calculus, physics, or engineering problems involving circular motion or rotational quantities.
Is there a difference between radians and degrees in trigonometric functions?: Yes, the trigonometric functions (sine, cosine, tangent) produce different results when using radians versus degrees. Most programming languages and calculators default to radians for these functions.
Can I convert radians to degrees without using π?: No, the conversion formulas require π because radians are defined based on the radius of a circle, which is π times the diameter.