Degrees to Radians: Understanding Angle Measurements

Angle measurement is fundamental in mathematics, engineering, navigation, and many other fields. While most people are familiar with degrees, radians and gradians serve important purposes in different contexts.

Degrees

The most familiar angle unit. A full circle is 360 degrees. This system dates back to ancient Babylonian mathematics, which used a base-60 number system.

Common angles:

• Right angle: 90°
• Straight angle: 180°
• Full rotation: 360°
• 45° angle (half of right angle)
• 60° angle (equilateral triangle)

Radians

Radians are the standard unit in mathematics and physics. One radian is the angle subtended by an arc equal in length to the radius of the circle.

• A full circle = 2π radians ≈ 6.283 radians
• π radians = 180°
• 1 radian ≈ 57.296°

Radians simplify many mathematical formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians.

Gradians (Gon)

Gradians divide a right angle into 100 equal parts, making a full circle 400 gradians. They are used in surveying and some European countries.

• Right angle = 100 grad
• Full circle = 400 grad
• 1 grad = 0.9°

Conversion Formulas

• Degrees to Radians: rad = deg × (π/180)
• Radians to Degrees: deg = rad × (180/π)
• Degrees to Gradians: grad = deg × (10/9)
• Gradians to Degrees: deg = grad × (9/10)

Quick Reference

Degrees	Radians	Gradians
0°	0	0
30°	π/6 ≈ 0.524	33.33
45°	π/4 ≈ 0.785	50
60°	π/3 ≈ 1.047	66.67
90°	π/2 ≈ 1.571	100
180°	π ≈ 3.142	200
360°	2π ≈ 6.283	400