Degrees to Radians Calculator with Pi Fraction

Convert degrees to radians with precise pi fraction results. This calculator provides exact conversions and explains the relationship between degrees and radians, including how to express results as fractions of π.

How to Use This Calculator

Enter the angle in degrees in the input field. The calculator will convert it to radians and display the result in both decimal and pi fraction forms. You can also visualize the conversion with the interactive chart.

Tip: For common angles like 30°, 45°, 60°, 90°, 180°, and 360°, the calculator will show exact pi fractions.

Degrees to Radians Conversion

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

Radians = Degrees × (π/180)

This conversion is based on the fact that a full circle is 360° or 2π radians. Therefore, one radian equals 180/π degrees.

Common degree-to-radian conversions include:

0° = 0 radians
30° = π/6 radians
45° = π/4 radians
60° = π/3 radians
90° = π/2 radians
180° = π radians
270° = 3π/2 radians
360° = 2π radians

Understanding Pi Fractions

Pi fractions express angles as fractions of π. This is particularly useful for common angles that divide the circle into equal parts. For example:

π/6 = 30°
π/4 = 45°
π/3 = 60°
π/2 = 90°
π = 180°
3π/2 = 270°
2π = 360°

Pi fractions provide exact representations of these angles without decimal approximations.

Worked Examples

Example 1: Convert 45° to Radians

Using the formula:

Radians = 45 × (π/180) = π/4

So, 45° is equal to π/4 radians.

Example 2: Convert 120° to Radians

Using the formula:

Radians = 120 × (π/180) = 2π/3

So, 120° is equal to 2π/3 radians.

Example 3: Convert 225° to Radians

Using the formula:

Radians = 225 × (π/180) = 5π/4

So, 225° is equal to 5π/4 radians.

Frequently Asked Questions

Why do we need to convert degrees to radians?

Degrees and radians are both units of angular measurement, but radians are the standard unit in many areas of mathematics, physics, and engineering. Radians provide a more natural way to express angles in trigonometric functions and calculus.

What is the difference between π/2 and 1.5708 radians?

π/2 is the exact value of 90° in radians, while 1.5708 is an approximation of π/2. The exact pi fraction is more precise and preferred in mathematical contexts.

Can I convert radians back to degrees?

Yes, you can use the inverse formula: Degrees = Radians × (180/π). Our radians to degrees calculator can handle this conversion as well.

What is the smallest angle that can be expressed as a simple pi fraction?

The smallest angle that can be expressed as a simple pi fraction is π/12 (15°), which is one-twelfth of a full circle.