Quick Answer

90 degrees equals approximately 1.5708 radians (exactly pi/2). One radian equals approximately 57.2958 degrees.

Degrees to Radians

Degrees

Radians to Degrees

Radians

Common Examples

Input	Result
45 degrees	0.7854 radians (pi/4)
180 degrees	3.1416 radians (pi)
360 degrees	6.2832 radians (2*pi)
1 radian	57.2958 degrees
2.5 radians	143.2394 degrees

How It Works

The Formula

The conversion between degrees and radians is based on the fact that a full circle is 360 degrees or 2pi radians. This gives the fundamental relationship:

pi radians = 180 degrees

From this, two conversion formulas follow:

Degrees to Radians: Radians = Degrees x (pi / 180)

Radians to Degrees: Degrees = Radians x (180 / pi)

Why Two Systems Exist

Degrees divide a circle into 360 equal parts, a convention dating back to Babylonian astronomy. Radians measure angles based on the radius of a circle: one radian is the angle subtended by an arc equal in length to the radius. Radians are the standard unit in calculus, physics, and most mathematical functions because they simplify many formulas. For example, the derivative of sin(x) equals cos(x) only when x is measured in radians.

Common Reference Angles

• 0 degrees = 0 radians
• 30 degrees = pi/6 radians (approximately 0.5236)
• 45 degrees = pi/4 radians (approximately 0.7854)
• 60 degrees = pi/3 radians (approximately 1.0472)
• 90 degrees = pi/2 radians (approximately 1.5708)
• 180 degrees = pi radians (approximately 3.1416)
• 360 degrees = 2pi radians (approximately 6.2832)

Worked Example

To convert 45 degrees to radians: 45 x (pi / 180) = 45pi / 180 = pi/4 = 0.7854 radians. To convert 2.5 radians to degrees: 2.5 x (180 / pi) = 450 / pi = 143.2394 degrees. To convert 270 degrees to radians: 270 x (pi / 180) = 270pi / 180 = 3pi/2 = 4.7124 radians.

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Frequently Asked Questions

Why do mathematicians prefer radians over degrees?

Radians simplify many mathematical formulas. In calculus, the derivative of sin(x) is cos(x) only when x is in radians. Taylor series expansions, Euler's formula (e^(ix) = cos(x) + i*sin(x)), and many physics equations all assume radian measure. Degrees require extra conversion factors in these contexts.

What is one radian in degrees?

One radian equals approximately 57.2958 degrees. It is the angle formed when the arc length equals the radius of the circle. This is not a round number in degrees because radians are based on pi, which is irrational.

How many radians are in a full circle?

A full circle contains 2pi radians, which is approximately 6.2832 radians. This follows from the circumference formula C = 2*pi*r: the full circumference subtends an angle equal to the circumference divided by the radius, which is 2*pi.

Can angles be negative?

Yes. Negative angles represent rotation in the clockwise direction, while positive angles represent counterclockwise rotation. This convention is standard in mathematics and physics. For example, -90 degrees equals -pi/2 radians.

What are gradians and how do they relate to degrees and radians?

Gradians (also called gons) divide a right angle into 100 parts, making a full circle 400 gradians. They are occasionally used in surveying and some European countries. To convert: 1 gradian = 0.9 degrees = pi/200 radians.