Radians vs. Degrees: A Complete Guide to Angle Units

When you first start learning geometry, everything is in degrees. A circle is 360°, a right angle is 90°. It's intuitive and familiar.

But as you advance into trigonometry, calculus, or physics, suddenly radians appear. Why make things complicated? Why do we need a second unit for the same thing?

This guide breaks down the differences, explains why radians are essential for higher math, and shows you how to convert between them effortlessly.

What is a Degree? (The "Human" Unit)

A degree (symbol: °) is a unit of angle measurement defined by dividing a full circle into 360 equal parts.

• Origin: Ancient Babylonians, who used a base-60 number system and estimated the year to be 360 days. (Learn more about the history of navigation angles).
• Best for: Navigation, construction, surveying, and everyday life.
• Key Values:
  • Full Circle: 360°
  • Straight Line: 180°
  • Right Angle: 90°

Why we love it: 360 is a great number because it can be divided evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, etc. It makes mental math easy.

What is a Radian? (The "Mathematical" Unit)

A radian (symbol: rad) is defined using the radius of the circle.

Definition: One radian is the angle created when the arc length is equal to the radius of the circle.

Imagine taking the radius of a circle, bending it into a curve, and laying it along the edge of the circle. The angle that arc covers is exactly 1 radian.

• Origin: Pure mathematics. It relates the angle directly to the physical properties of the circle (radius and circumference).
• Best for: Calculus, physics, computer programming (most code libraries use radians).
• Key Values:
  • Full Circle: 2π radians (≈ 6.28 rad)
  • Straight Line: π radians (≈ 3.14 rad)
  • Right Angle: π/2 radians (≈ 1.57 rad)

The Conversion Formula

Converting between the two is simple if you remember one fact: 180° = π radians

Degrees to Radians

Multiply by π / 180

Radians = Degrees × π / 180

Example: Convert 45° to radians. 45 × π / 180 = 45π / 180 = π / 4 radians

Radians to Degrees

Multiply by 180 / π

Degrees = Radians × 180 / π

Example: Convert π / 3 radians to degrees. π / 3 × 180 / π = 180 / 3 = 60°

Why Do Mathematicians Prefer Radians?

In advanced math (Calculus), formulas become much simpler with radians.

For example, if x is in radians:

• The derivative of sin(x) is simply cos(x).

If x were in degrees, the derivative would be:

• π / 180 cos(x)

That extra factor of π / 180 would appear in every single calculation, making equations messy and complicated. Radians are the "natural" unit for mathematics.

Summary Table

| Angle | Degrees | Radians (Exact) | Radians (Decimal) | | :--- | :--- | :--- | :--- | | Zero | 0° | 0 | 0 | | 30° | 30° | π/6 | 0.52 | | 45° | 45° | π/4 | 0.79 | | 60° | 60° | π/3 | 1.05 | | Right Angle | 90° | π/2 | 1.57 | | Straight | 180° | π | 3.14 | | Full Circle | 360° | 2π | 6.28 |

Need to convert angles quickly? Use our online tool which supports both degree and radian inputs.