How do you convert degrees to radians?

To convert degrees to radians, multiply the degree value by pi divided by 180. The formula is: radians = degrees x (pi / 180). For example, 90 degrees equals 90 x (pi / 180) = pi / 2 radians, which is approximately 1.5708 radians. A full circle is 360 degrees or 2 x pi radians (approximately 6.2832 radians).

What is a gradian (grad)?

A gradian (also called grad or gon) is an angle unit where a full circle equals 400 gradians. This makes right angles exactly 100 gradians, which was designed to simplify decimal calculation. Gradians are mainly used in surveying and some European engineering fields. To convert degrees to gradians, multiply by 400/360 (or 10/9). For example, 90 degrees equals 100 gradians.

What is the difference between arc minutes and arc seconds?

Arc minutes and arc seconds are subdivisions of a degree used for precise angle measurement. One degree equals 60 arc minutes (written as 60'). One arc minute equals 60 arc seconds (written as 60''). So one degree equals 3,600 arc seconds. These units are commonly used in navigation, astronomy, and geographic coordinate systems (latitude and longitude). For example, the angle 1 degree 30 arc minutes equals 1.5 degrees.

Angle Conversion Calculator

Convert any angle between degrees, radians, gradians, turns, arc minutes, and arc seconds. Enter a value in any unit and see all equivalent measurements instantly.

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Standard Exact mathematical conversions. No measurement uncertainty.

1. Enter Angle

Input unit

Angle value

Please enter a valid number.

Common angles (click to fill)

2. Conversion Results

Degrees

Radians

1.5708

Gradians

100

Turns

0.25

Arc Minutes

5,400

Arc Seconds

324,000

Full Conversion Summary

Degrees
90
deg

Radians

1.5708

rad

Gradians

100

grad

Turns

0.25

full rotations

Arc Minutes

5,400

' (arc min)

Arc Seconds

324,000

'' (arc sec)

Worked Example (default: 90 degrees)

Input: 90 degrees

Radians = 90 x (pi / 180) = pi / 2 = 1.570796...

Gradians = 90 x (400 / 360) = 90 x (10/9) = 100

Turns = 90 / 360 = 0.25 (a quarter circle)

Arc minutes = 90 x 60 = 5,400'

Arc seconds = 90 x 3,600 = 324,000''

Common Angle Reference Table

Angle	Degrees	Radians	Gradians	Turns	Arc Seconds
Zero	0	0	0	0	0
30 degrees (1/12 circle)	30	pi/6 ≈ 0.5236	33.33	0.0833	108,000
45 degrees (1/8 circle)	45	pi/4 ≈ 0.7854	50	0.125	162,000
60 degrees (1/6 circle)	60	pi/3 ≈ 1.0472	66.67	0.1667	216,000
Right angle (1/4 circle)	90	pi/2 ≈ 1.5708	100	0.25	324,000
Straight angle (1/2 circle)	180	pi ≈ 3.1416	200	0.5	648,000
3/4 circle	270	3pi/2 ≈ 4.7124	300	0.75	972,000
Full circle	360	2pi ≈ 6.2832	400	1	1,296,000

Angle Units Explained

An angle describes the amount of rotation between two rays sharing a common endpoint. Several different units are used to measure angles, each suited to particular fields and applications.

Degrees

The degree is the most familiar unit of angle measurement. A full circle is divided into 360 degrees. This choice of 360 dates from ancient Babylon and was convenient because 360 has many divisors (it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180). Degrees are used in everyday navigation, geography, carpentry, and most applied contexts.

Radians

Radians are the standard unit of angle measurement in mathematics and physics. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. A full circle is 2 x pi radians (approximately 6.2832). Radians are the natural unit for trigonometric functions and calculus because they simplify derivatives: d/dx(sin x) = cos x only holds when x is in radians. To convert degrees to radians: multiply by pi / 180. To convert radians to degrees: multiply by 180 / pi.

Gradians (Grads)

A gradian (also called grad or gon) divides the full circle into 400 equal parts. A right angle is exactly 100 gradians, a straight angle is 200, and a full circle is 400. Gradians were introduced during the French metric reform to facilitate decimal calculation. They are used in surveying, civil engineering, and some European technical fields. The conversion is: gradians = degrees x (400 / 360) = degrees x (10 / 9).

Turns

A turn (also called revolution or cycle) represents one complete rotation: 360 degrees, 2 x pi radians, or 400 gradians. Expressing angles as a fraction of a full turn is intuitive for rotational contexts such as gears, propellers, and clocks. For example, a quarter turn is 90 degrees and half a turn is 180 degrees.

Arc Minutes and Arc Seconds

Arc minutes and arc seconds subdivide the degree for high-precision angle measurements. One degree equals 60 arc minutes (symbol: '). One arc minute equals 60 arc seconds (symbol: ''). So one degree equals 3,600 arc seconds. These subdivisions are used in navigation, astronomy, cartography, and geographic coordinate systems. For example, a position given as 36 degrees 30 minutes south latitude equals 36.5 degrees south.

Conversion Formulas

From	To	Formula
Degrees	Radians	rad = deg x (pi / 180)
Radians	Degrees	deg = rad x (180 / pi)
Degrees	Gradians	grad = deg x (10 / 9)
Gradians	Degrees	deg = grad x (9 / 10)
Degrees	Turns	turns = deg / 360
Turns	Degrees	deg = turns x 360
Degrees	Arc minutes	arcmin = deg x 60
Degrees	Arc seconds	arcsec = deg x 3600
Arc minutes	Degrees	deg = arcmin / 60
Arc seconds	Degrees	deg = arcsec / 3600

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Sources and method: Conversion factors are exact mathematical relationships: 1 full circle = 360 degrees = 2 x pi radians = 400 gradians = 1 turn = 21,600 arc minutes = 1,296,000 arc seconds. These are definitional constants with no measurement uncertainty. Reference: ISO 80000-3 Quantities and units (Space and time).

This calculator performs exact mathematical conversions. Results beyond approximately 15 significant figures may show floating-point rounding inherent to all digital computation. For scientific work requiring exact rational expressions (such as pi/2 rather than 1.5708), note the exact form in the reference table above.