Radians to Degrees Converter

Convert angles from radians to degrees with precision. Visual representation, unit circle visualization, and trigonometric values included.

Conversion Formula:

Degrees = Radians × 180/π

Where π ≈ 3.141592653589793

rad
Enter an angle in radians (π = 3.141592653589793)
Number of decimal places in the result
Format for displaying the result
0 rad
π/6 rad
π/4 rad
π/3 rad
π/2 rad
π rad
3π/2 rad
2π rad
1 rad
30°
45°
60°
90°
180°
360°
57.2958°
Enter a π fraction like "π/2" or "3π/4". Click the button to use it.
Converting...

Understanding Radians and Degrees

Radians and degrees are two units for measuring angles. Radians are based on the radius of a circle, while degrees divide a circle into 360 equal parts.

Key Mathematical Relationships:

π radians = 180°

2π radians = 360°

1 radian = 180/π degrees ≈ 57.2958°

1° = π/180 radians ≈ 0.0174533 radians

Why Convert Radians to Degrees?

1

Practical Applications: Degrees are commonly used in navigation, geography, construction, and everyday measurements where intuitive understanding is important.

2

Educational Context: Degrees are often taught first in mathematics education, making conversion necessary when working with radians in advanced mathematics.

3

Interdisciplinary Work: When collaborating across fields (e.g., mathematicians working with engineers or designers), conversion ensures clear communication.

Common Radian to Degree Conversions

Radians (π) Radians (decimal) Degrees
0 0.0000 0°
π/6 0.5236 30°
π/4 0.7854 45°
π/3 1.0472 60°
π/2 1.5708 90°
π 3.1416 180°
3π/2 4.7124 270°
2π 6.2832 360°

Visualizing Radians

Definition of a Radian:

One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

Arc Length = Radius × Angle (in radians)

1

Full Circle: The circumference of a circle is 2πr, so a full circle contains 2π radians (approximately 6.2832 radians).

2

Right Angle: A right angle (90°) is π/2 radians (approximately 1.5708 radians).

3

Straight Angle: A straight angle (180°) is π radians (approximately 3.1416 radians).

Applications of Radian to Degree Conversion

  • Trigonometry: Converting between radian measures in trigonometric functions and degree measures in practical applications
  • Physics: Converting angular velocity from rad/s to °/s for intuitive understanding
  • Engineering: Interpreting angular measurements in mechanical design and analysis
  • Computer Graphics: Converting between radian-based mathematical calculations and degree-based user interfaces
  • Navigation: Converting bearing angles from mathematical calculations to navigational degrees

Calculator Features:

  • Precise conversion from radians to degrees
  • Support for π fraction input (e.g., π/2, 3π/4)
  • Visual representation on a unit circle
  • Trigonometric function values (sin, cos, tan)
  • Multiple output formats: decimal degrees and DMS
  • Arc length calculation for unit circle
  • Common radian values for quick testing

Frequently Asked Questions

The relationship π radians = 180° comes from the definition of a radian. The circumference of a circle is 2πr, and a full circle is 360°. Since the arc length for a full circle is 2πr (when the angle is in radians), and this corresponds to 360°, we have 2π radians = 360°, which simplifies to π radians = 180°.

For a quick mental approximation, multiply radians by 57.3 (since 1 radian ≈ 57.2958°). For more accurate conversions, use the formula: Degrees = Radians × (180/π). Remember common conversions: π/2 ≈ 1.5708 radians = 90°, π ≈ 3.1416 radians = 180°, 2π ≈ 6.2832 radians = 360°.

Radians provide a natural way to measure angles based on the properties of circles. They simplify many mathematical formulas, particularly in calculus and physics. For example, the derivative of sin(x) is cos(x) only when x is measured in radians. Radians also provide a direct relationship between angular and linear measurements: arc length = radius × angle (in radians).

You can enter π fractions in the "Enter as π Fraction" field. Type expressions like "π/2", "3π/4", "2π", or even "π". The calculator will convert these to their decimal equivalents and perform the conversion. This is especially useful for exact mathematical expressions that involve π.

"Arc Length (r=1)" shows the length of the arc on a unit circle (radius = 1) corresponding to the given angle in radians. This demonstrates the definition of a radian: for a unit circle, the arc length equals the angle in radians. For example, an angle of 1.5708 radians (π/2) has an arc length of 1.5708 on a unit circle.