How do you convert degrees to radians?

Multiply the angle in degrees by pi/180. radians = degrees x pi / 180 Examples: - 180 degrees: 180 x pi/180 = pi = 3.14159265... - 90 degrees: 90 x pi/180 = pi/2 = 1.57079633 - 45 degrees: 45 x pi/180 = pi/4 = 0.78539816 - 30 degrees: 30 x pi/180 = pi/6 = 0.52359878 - 60 degrees: 60 x pi/180 = pi/3 = 1.04719755 - 1 degree: pi/180 = 0.01745329 radians Memory trick: divide the angle by 180 and multiply by pi. Or remember the key fact that 180 degrees = pi radians, then scale proportionally.

How do you convert radians to degrees?

Multiply the angle in radians by 180/pi. degrees = radians x 180 / pi Examples: - pi radians: pi x 180/pi = 180 degrees - pi/2 radians: (pi/2) x 180/pi = 90 degrees - 1 radian: 1 x 180/pi = 57.2957795 degrees - 2.5 radians: 2.5 x 180/pi = 143.2394 degrees - 2*pi radians: 2*pi x 180/pi = 360 degrees Note: 1 radian is approximately 57.3 degrees — a useful approximation to remember. One radian is the angle where the arc length equals the radius.

Why do we use radians instead of degrees?

Radians are mathematically natural in a way that degrees are not. Degrees are arbitrary (the choice of 360 comes from ancient Babylonian astronomy). Radians are defined geometrically: 1 radian is the angle subtended when the arc length equals the radius. Why radians are essential in mathematics: - Calculus: the derivative of sin(x) is cos(x) ONLY when x is in radians. In degrees: d/dx sin(x degrees) = (pi/180) cos(x degrees). - Taylor series: sin(x) = x - x^3/6 + x^5/120 - ... only works in radians. - Complex numbers: e^(i*theta) = cos(theta) + i*sin(theta) requires radians. - Arc length: s = r*theta (simple formula) requires theta in radians. In programming: virtually all math libraries (Python math, JavaScript Math, C math.h) use radians for trig functions. Always convert to radians before calling sin(), cos(), tan().

What are gradians and when are they used?

Gradians (also called grads or gon) divide a full circle into 400 parts instead of 360 degrees or 2*pi radians. A right angle = 100 gradians. Conversions: - 360 degrees = 400 gradians - 1 degree = 10/9 gradians (about 1.1111) - 1 gradian = 0.9 degrees - 90 degrees = 100 gradians - 180 degrees = 200 gradians Where gradians are used: - Surveying and civil engineering (some European countries and older equipment) - Military angle measurement (some contexts) - Some scientific calculators have a GRAD mode alongside DEG and RAD The appeal is that a right angle is a round number (100 grad) and slopes can be expressed cleanly. However degrees and radians dominate globally, and gradians are rarely encountered in everyday mathematics or programming.

What is the relationship between pi and a full circle?

A full circle is exactly 2*pi radians. Pi (pi = 3.14159265...) is defined as the ratio of a circle's circumference to its diameter: C = pi*d = 2*pi*r. Key circle facts using radians: - Full circle: 2*pi radians = 360 degrees - Half circle: pi radians = 180 degrees - Quarter circle: pi/2 radians = 90 degrees - Arc length: s = r*theta (theta in radians) - Area of sector: A = (1/2)*r^2*theta (theta in radians) Why pi is irrational: pi cannot be written as a fraction p/q of two integers. Its decimal expansion never terminates or repeats: 3.14159265358979323846... This means most angles in degrees have non-terminating radian equivalents. The clean exceptions are 0, 30, 45, 60, 90, 120, 135, 150, 180, and their multiples -- these are the standard angles in trigonometry.

Degrees to Radians Converter — Bidirectional Angle Converter

The Degrees to Radians Converter converts any angle between degrees, radians, and gradians instantly. Fully bidirectional: switch between degrees-to-radians and radians-to-degrees with one click, and the previous result is automatically loaded as the new input. Exact pi expressio...

ANGLE CONVERTER

Angle in Degrees (°)

RADIANS

3.14159265

= π

DEGREES

180°

RADIANS

3.14159265

GRADIANS

200

Turns (full rotations)

0.5

Quadrant

Gradians

200 grad

Exact π form

Given: 180°

Formula: radians = degrees × π / 180

= 180 × π / 180

= 180 × 0.01745329...

= 3.1415926536 rad (= π)

Gradians: 180 / 0.9 = 200 grad

Turns: 180 / 360 = 0.5 turns

QUICK EXAMPLES (degrees)

UNIT CIRCLE POSITION

Angle shown on the unit circle

COMMON ANGLE REFERENCE

Degrees	Radians	Exact	Gradians
0°	0	0	0
30°	0.523599	π/6	33.33̄
45°	0.785398	π/4	50
60°	1.047198	π/3	66.67
90°	1.570796	π/2	100
120°	2.094395	2π/3	133.33̄
135°	2.356194	3π/4	150
150°	2.617994	5π/6	166.67
180°	3.141593	π	200
270°	4.712389	3π/2	300
360°	6.283185	2π	400

Click any row to load that angle

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FORMULAS

ANGLE CONVERSION FORMULAS

NAME	FORMULA	DESCRIPTION
Degrees to radians	radians = degrees x pi / 180	Multiply degrees by pi/180 (approx 0.01745)
Radians to degrees	degrees = radians x 180 / pi	Multiply radians by 180/pi (approx 57.296)
Degrees to gradians	gradians = degrees / 0.9	400 gradians = 360 degrees
Radians to gradians	gradians = radians x 200 / pi	400 gradians = 2*pi radians
Turns	turns = degrees / 360	Full rotations (1 turn = 360 degrees = 2*pi rad = 400 grad)
1 radian in degrees	1 rad = 180/pi = 57.29577951 degrees	One radian expressed as a degree value
1 degree in radians	1 degree = pi/180 = 0.01745329 rad	One degree expressed as a radian value
Key conversions	30=pi/6, 45=pi/4, 60=pi/3, 90=pi/2, 180=pi, 360=2pi	Most commonly needed exact conversions

HOW TO USE

1
Select your conversion direction: Degrees to Radians or Radians to Degrees. The direction button also swaps your previous result into the input, so you can convert back and forth seamlessly.
2
Enter your angle value. For degrees, try whole numbers like 30, 45, 60, 90, or 180. For radians, try values like 1, 1.5708 (pi/2), or 3.14159 (pi). Use the quick example buttons for common angles.
3
The result appears in the large black box showing the converted value, the exact pi expression where applicable (e.g. pi/6 for 30 degrees), and all three units simultaneously: degrees, radians, and gradians.
4
The unit circle diagram shows the angle position. The angle arc sweeps from 0 to your angle, and the point on the circle is marked with coordinates. The quadrant is labelled.
5
The common angles table on the right is clickable — click any row to load that angle instantly. The current angle is highlighted in black so you can see where you are in the standard angle list.

WORKED EXAMPLE

180 deg to rad: 180 x pi/180 = pi = 3.14159265. 45 deg = pi/4 = 0.78539816 rad = 50 grad. 90 deg = pi/2 = 1.57079633 rad = 100 grad. Reverse: 1 rad x 180/pi = 57.29577951 deg. 2.5 rad = 143.2394 deg. 360 deg = 2pi rad = 400 grad = 1 turn.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.