Degrees and Angles Calculator

Convert between degrees, radians, and gradians with our free online calculator. Learn about angle units, conversions, and practical applications.

What are angle units?

Angles are measured in different units depending on the context. The three most common units are degrees, radians, and gradians. Each has its own advantages and is used in different fields.

Degrees (°) are the most common unit for measuring angles in everyday life. A full circle is 360 degrees, making them intuitive for basic measurements.

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

Gradians (grad) are used in surveying and navigation. A full circle is 400 gradians, making calculations in these fields more straightforward.

Conversion formulas

Converting between angle units is straightforward using these formulas:

Degrees to Radians: radians = degrees × (π/180)

Degrees to Gradians: gradians = degrees × (400/360)

Radians to Degrees: degrees = radians × (180/π)

Radians to Gradians: gradians = radians × (400/(2π))

Gradians to Degrees: degrees = gradians × (360/400)

Gradians to Radians: radians = gradians × (2π/400)

Where π (pi) is approximately 3.141592653589793.

Practical applications

Angle conversions are essential in various fields:

Mathematics: Trigonometry and calculus use radians.
Physics: Angular momentum and wave theory use radians.
Engineering: Mechanical and electrical systems often use degrees.
Navigation: Gradians are used in surveying and mapping.
Computer Graphics: Degrees are common for rotations.

Understanding these conversions helps professionals work across different disciplines.

Common angle conversions

Here are some frequently used angle conversions:

Degrees	Radians	Gradians
0°	0 rad	0 grad
30°	0.5236 rad	33.3333 grad
45°	0.7854 rad	50 grad
60°	1.0472 rad	66.6667 grad
90°	1.5708 rad	100 grad
180°	3.1416 rad	200 grad
270°	4.7124 rad	300 grad
360°	6.2832 rad	400 grad

FAQ

What is the difference between degrees and radians?

Degrees are based on a 360-degree circle, while radians are based on the radius of a circle. One radian is the angle where the arc length equals the radius. Degrees are more intuitive for everyday use, while radians are standard in advanced mathematics and physics.

When should I use gradians instead of degrees?

Gradians are primarily used in surveying and navigation because they simplify calculations involving right angles. A full circle is 400 gradians, making conversions for 90-degree angles straightforward.

How do I convert between degrees and radians?

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 90° is equal to π/2 radians (approximately 1.5708).

What are some common angle measurements?

Common angle measurements include 0° (no angle), 30° (common in triangles), 45° (diagonal of a square), 60° (equilateral triangle), 90° (right angle), 180° (straight line), and 360° (full circle).

Why are radians used in calculus?

Radians simplify calculus because the derivative of sin(x) is cos(x) when x is in radians. This makes trigonometric functions more compatible with calculus operations like integration and differentiation.