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Angle converter

Convert degrees to radians and radians to degrees, plus gradians, arcminutes, arcseconds, turns and milliradians, with π carried to 50 digits.

Updated Checked against 5 worked examples

A number in the “from” unit, or with its own unit
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Converted value
rad
Converted value: 1.57079632679 rad
Shown to 12 significant figures, half-up
Conversion
90° ≈ 1.570796 rad
Conversion factor
0.0174532925199
Exactness
Exact definitions, involves π

90° is about 1.570796 rad. One degree is about 0.01745329 rad.

The angle

90°90° ≈ 1.570796 rad
90° in every angle unit (7 rows)
UnitValueDefinition
radian (rad)1.570796Base unit
degree (°)901° = π/180 rad, exact
gradian (gon)1001 gon = π/200 rad, exact
arcminute (′)5,4001′ = π/10,800 rad, exact
arcsecond (″)324,0001″ = π/648,000 rad, exact
turn0.251 turn = 2π rad, exact
milliradian (mrad)1,570.7961 mrad = 0.001 rad, exact
How it's calculated S

  1. Convert degree to radian

    90°×π180 rad°=1.57079632679 rad90\text{°} \times \tfrac{\pi}{180}\ \tfrac{\text{rad}}{\text{°}} = 1.57079632679\ \text{rad}

    1° = π/180 rad, exact. The units are defined exactly in terms of π, so the result is irrational; it is shown to 12 significant figures.

About the angle converter

The radian is the SI unit of angle: the angle whose arc equals the radius, so a full turn is 2π rad. Degrees divide the turn into 360 and gradians (gon) into 400, and arcminutes and arcseconds are 1/60 and 1/3,600 of a degree. Converting degrees to radians multiplies by π/180, so the result is irrational and is shown to 12 significant figures.

The default converts a right angle: 90° = π/2 = 1.57079632679 rad. Programmers need this because the trigonometric functions of most languages take radians, and surveyors and astronomers move between degrees, gradians and arcseconds.

Conversions among degrees, arcminutes, arcseconds, gradians and turns never involve π and are exact: 1 turn = 360° = 400 gon. The milliradian here is the true 0.001 rad, about 6,283.2 to a turn.

Worked examples

90° in radians

Value
90 °
From
degree (°)
To
radian (rad)
Converted value
1.5707963268 rad
Exactness
Exact definitions, involves π

Checked against: 90 × π/180 = π/2 = 1.57079632679489661923… (π to 60 digits in Python decimal)

1 turn in degrees

Value
1 turn
From
turn
To
degree (°)
Converted value
360 °
Exactness
Exact

Checked against: 1 turn = 2π rad = 360° (π cancels)

1° in arcseconds

Value
1 °
From
degree (°)
To
arcsecond (″)
Converted value
3,600 ″

Checked against: 1° = 60′ = 3600″ (SI Brochure Table 8)

1 radian in degrees

Value
1 rad
From
radian (rad)
To
degree (°)
Converted value
57.2957795131 °

Checked against: 180/π = 57.29577951308232087… (Python decimal, 60 digits)

Questions

How do I convert degrees to radians?

Multiply by π/180, about 0.0174533. 90° is π/2 = 1.5708 rad, 180° is π = 3.14159 rad and 45° is π/4 = 0.785398 rad. To go back, multiply radians by 180/π, so 1 rad is 57.2958°. JavaScript's Math.sin and Python's math.sin both expect radians.

How many degrees are in a radian?

57.2957795°, which is 180/π. A radian is the angle at the centre of a circle cut off by an arc as long as the radius; a full circumference is 2π radii, so a full turn is 2π rad = 360°. The SI Brochure defines the radian as 1 m/m, a coherent SI unit.

What is a gradian?

One 400th of a full turn, so a right angle is exactly 100 gradians (symbol gon, also called grads). One gon is 0.9° or 54 arcminutes. It is used mainly in surveying, where a quarter circle of 100 keeps slopes and bearings decimal, and calculators label the mode GRAD.

Is a mil the same as a milliradian?

Not always. A true milliradian is 0.001 rad, about 6,283.2 per turn, and that is the unit used here. NATO compasses and artillery use a mil of 1/6,400 turn, 0.98175 mrad, chosen so the circle divides evenly. Rifle scopes marked MRAD use the true 0.001 rad, where 1 mrad spans 10 cm at 100 m.

How many arcseconds are in a degree?

3,600: a degree is 60 arcminutes (′) and an arcminute is 60 arcseconds (″), as listed in Table 8 of the SI Brochure. One arcsecond of latitude is about 30.9 m on the ground, because the nautical mile, 1,852 m, was based on one arcminute of latitude.

How accurate is the angle converter?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “90° in radians” is checked against 90 × π/180 = π/2 = 1.57079632679489661923… (π to 60 digits in Python decimal).

Where does the method come from?

NIST SP 811 (2008), Guide for the Use of the SI, Appendix B — conversion factors; NIST Handbook 44, Appendix C — General tables of units of measurement; BIPM SI Brochure, 9th edition (2019), Table 8 — degree, minute, second of arc.

About this calculator

xto=xfrom×ffromftof=one unit expressed in rad\begin{gathered} x_{\text{to}} = x_{\text{from}} \times \dfrac{f_{\text{from}}}{f_{\text{to}}} \\ f = \text{one unit expressed in } \text{rad} \end{gathered}

Sources

  1. NIST SP 811 (2008), Guide for the Use of the SI, Appendix B — conversion factors
  2. NIST Handbook 44, Appendix C — General tables of units of measurement
  3. BIPM SI Brochure, 9th edition (2019), Table 8 — degree, minute, second of arc

Checked against references

5 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.

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