Degrees to Radians Converter
Convert degrees to radians (° to rad) with exact π fractions
1 degree = π/180 radians ≈ 0.0174533 rad. To convert degrees to radians, multiply by π/180, so 90° = π/2 ≈ 1.5708 rad.
Degrees to Radians converter
Live- 1.1111111 gon
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- 0.0027777778 rev
How to convert degrees to radians
A full turn is 360 degrees, and it is also 2π radians, because a circle's circumference is 2π times its radius and a radian is the angle that cuts off one radius of arc. Setting the two equal gives 360° = 2π rad, so 1° = π/180 rad. Multiplying any angle in degrees by π/180 converts it to radians. The radians to degrees converter undoes it with 180/π, and the angle converter shows gradians, turns and arcminutes as well.
There are two ways to write the answer. The decimal form, such as 0.785398 rad, is what calculators and code use. The exact form keeps π as a symbol: 45° × π/180 = 45π/180, which reduces to π/4. Textbooks and exams usually want the exact form, so reduce the fraction instead of reaching for decimals.
Step by step
- Take the angle in degrees.
- Multiply it by π, then divide by 180 (or multiply by 0.01745329).
- For an exact answer, reduce the fraction and keep π, so 60° becomes π/3.
See it on a scale
Worked examples
Three angles from trigonometry homework and engineering drawings: 30°, 45° and a three-quarter turn of 270°.
30 ° ÷ 57.29577951 = 0.52359878 rad
= 0.52359878 rad
45 ° ÷ 57.29577951 = 0.78539816 rad
= 0.78539816 rad
270 ° ÷ 57.29577951 = 4.712389 rad
= 4.712389 rad
How do I convert degrees, minutes and seconds to radians?
Convert to decimal degrees, then multiply by π/180: 40° 26′ 46″ is 40 + 26/60 + 46/3600 = 40.446111°, which is 0.705918 rad.
This comes up whenever map coordinates feed a formula. Latitudes and longitudes are often written in degrees, minutes and seconds, but distance formulas such as the haversine formula, and the trigonometric functions of every programming language, need radians.
- Add minutes ÷ 60 and seconds ÷ 3,600 to the whole degrees.
- Make the value negative for south latitudes and west longitudes.
- Multiply by π/180, or 0.01745329.
A longitude of 79° 58′ 56″ W therefore becomes −79.982222°, or −1.395953 rad. Keep at least six decimals in radians: on the Earth's surface, 0.000001 rad spans about 6.4 m. With the mean Earth radius of 6,371 km, the haversine result comes out in kilometers, and the km to miles converter finishes the job.
How do you convert negative angles or angles over 360° to radians?
Use the same π/180 factor, sign and all: −45° is −π/4 rad (−0.785398), and 450° is 5π/2 rad (7.853982).
Angles beyond a full turn point the same way as a smaller angle, so you can reduce them before or after converting. Add or subtract 360°, or 2π rad, until the angle lands in the range you need:
| Degrees | Radians (exact) | Radians (decimal) | Same direction as |
|---|---|---|---|
| −90° | −π/2 | −1.570796 | 270° (3π/2) |
| −45° | −π/4 | −0.785398 | 315° (7π/4) |
| 450° | 5π/2 | 7.853982 | 90° (π/2) |
| 720° | 4π | 12.566371 | 0° (0) |
| 1,000° | 50π/9 | 17.453293 | 280° (14π/9) |
Keep the sign when it carries meaning. In motor control and robotics, −90° and 270° finish at the same position but turn in opposite directions, so the software may treat them differently.
Degrees to radians conversion chart
Degrees converted to radians and rounded to six decimal places. For the exact π form of the unit-circle angles, see the next table.
| Degrees (°) | Radians (rad) |
|---|---|
| 1 ° | 0.017453 rad |
| 2 ° | 0.034907 rad |
| 5 ° | 0.087266 rad |
| 10 ° | 0.174533 rad |
| 15 ° | 0.261799 rad |
| 18 ° | 0.314159 rad |
| 20 ° | 0.349066 rad |
| 22.5 ° | 0.392699 rad |
| 30 ° | 0.523599 rad |
| 36 ° | 0.628319 rad |
| 40 ° | 0.698132 rad |
| 45 ° | 0.785398 rad |
| 50 ° | 0.872665 rad |
| 60 ° | 1.047198 rad |
| 72 ° | 1.256637 rad |
| 75 ° | 1.308997 rad |
| 90 ° | 1.570796 rad |
| 100 ° | 1.745329 rad |
| 120 ° | 2.094395 rad |
| 135 ° | 2.356194 rad |
| 150 ° | 2.617994 rad |
| 180 ° | 3.141593 rad |
| 210 ° | 3.665191 rad |
| 225 ° | 3.926991 rad |
| 240 ° | 4.18879 rad |
| 270 ° | 4.712389 rad |
| 300 ° | 5.235988 rad |
| 315 ° | 5.497787 rad |
| 330 ° | 5.759587 rad |
| 360 ° | 6.283185 rad |
| 720 ° | 12.566371 rad |
Degrees vs radians: what is the difference?
Degrees divide a turn into 360 equal parts by convention, while radians measure an angle by the arc it cuts on a circle, so one unit is chosen and the other comes from geometry.
| Attribute | Degrees | Radians |
|---|---|---|
| Symbol | ° | rad |
| System | Common | SI derived unit |
| Defined as | 1/360 of a turn | 180/π ° (≈ 57.29578°) |
| One unit equals | 0.017453293 rad | 57.29578 ° |
| Full turn | 360° | 2π rad ≈ 6.283185 |
| Status in the SI | Non-SI unit accepted for use with the SI | Coherent SI derived unit, equal to m/m = 1 |
| Where it is used | Navigation, maps, construction and school geometry | Calculus, physics, engineering and nearly all programming libraries |
| Exact form of common angles | Whole numbers (30°, 45°, 60°) | Fractions of π (π/6, π/4, π/3) |
4 key differences
- One radian is about 57.2958°, so the same angle is always a number about 57 times smaller in radians than in degrees.
- The radian is dimensionless, defined as the ratio m/m, yet the SI keeps the name rad so that angular speed in rad/s is not confused with frequency in hertz.
- Degrees subdivide into 60 arcminutes and 3,600 arcseconds, while radians subdivide decimally into milliradians and microradians.
- A 1° error is only 0.017453 rad, which is why radian values need more decimal places than degree values for the same precision.
Unit circle angles as π fractions
The standard angles of the unit circle in degrees and radians, with their sine and cosine rounded to four decimal places. The exact π fractions come from reducing degrees ÷ 180.
| Degrees | Radians (exact) | Radians (decimal) | sin | cos |
|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 1 |
| 30° | π/6 | 0.523599 | 0.5 | 0.866 |
| 45° | π/4 | 0.785398 | 0.7071 | 0.7071 |
| 60° | π/3 | 1.047198 | 0.866 | 0.5 |
| 90° | π/2 | 1.570796 | 1 | 0 |
| 120° | 2π/3 | 2.094395 | 0.866 | −0.5 |
| 135° | 3π/4 | 2.356194 | 0.7071 | −0.7071 |
| 150° | 5π/6 | 2.617994 | 0.5 | −0.866 |
| 180° | π | 3.141593 | 0 | −1 |
| 210° | 7π/6 | 3.665191 | −0.5 | −0.866 |
| 225° | 5π/4 | 3.926991 | −0.7071 | −0.7071 |
| 240° | 4π/3 | 4.18879 | −0.866 | −0.5 |
| 270° | 3π/2 | 4.712389 | −1 | 0 |
| 300° | 5π/3 | 5.235988 | −0.866 | 0.5 |
| 315° | 7π/4 | 5.497787 | −0.7071 | 0.7071 |
| 330° | 11π/6 | 5.759587 | −0.5 | 0.866 |
| 360° | 2π | 6.283185 | 0 | 1 |
The DEG / RAD calculator mistake
A scientific calculator does not know whether “30” means degrees or radians; it uses whatever mode it is set to. Type sin(30) expecting 0.5 while the calculator is in RAD mode and you get −0.98803, because it has taken the sine of 30 radians, an angle of about 1,719°. This single slip accounts for a large share of wrong answers in physics and trigonometry tests.
How to spot it
- sin(30) should be exactly 0.5. In RAD mode you get −0.988.
- cos(60) should be 0.5. In RAD mode you get −0.9524.
- tan(45) should be 1. In RAD mode you get 1.6198.
Look for a small D, DEG, R or RAD indicator on the display, and check it before every exam. If a problem gives angles in degrees, stay in DEG mode; if it gives expressions with π, switch to RAD. Some calculators also have a GRAD mode, covered in the degrees to gradians converter.
Degrees to radians in code and spreadsheets
Almost every programming language and spreadsheet computes sine, cosine and tangent in radians, so degree inputs must be converted first. These are the standard ways to do it.
| Environment | Degrees to radians | Example: sine of 30° |
|---|---|---|
| Excel and Google Sheets | RADIANS(x) or x*PI()/180 | =SIN(RADIANS(30)) |
| Python | math.radians(x) | math.sin(math.radians(30)) |
| NumPy | np.deg2rad(x) or np.radians(x) | np.sin(np.deg2rad(30)) |
| JavaScript | x * Math.PI / 180 | Math.sin(30 * Math.PI / 180) |
| Java | Math.toRadians(x) | Math.sin(Math.toRadians(30)) |
| C and C++ | x * M_PI / 180.0 | sin(30 * M_PI / 180.0) |
Why are radians the natural unit for angles?
Radians are natural because they measure an angle as arc length divided by radius, a pure ratio taken from the circle itself, whereas splitting a turn into 360 parts is a human convention. That is why formulas become simpler in radians:
- Arc length: s = rθ. A 60° arc on a circle of radius 10 cm is 10 × π/3 = 10.47 cm long.
- Sector area: A = ½r²θ. A 45° slice of a 12-inch pizza (6-inch radius) covers ½ × 36 × π/4 = 14.14 square inches.
- Small angles: for small θ in radians, sin θ ≈ θ. At 5°, θ = 0.08727 rad and sin θ = 0.08716, almost the same.
- Rotation speed: angular velocity is measured in radians per second. One revolution per minute is 2π/60 ≈ 0.10472 rad/s; the rpm to Hz converter handles the revolutions-per-second step.
Calculus is the strongest reason. The derivative of sin x is cos x only when x is in radians; in degrees an extra π/180 factor appears in every step.
How do you write an angle in terms of π?
Put the degrees over 180, reduce the fraction and attach π: 150° becomes 150/180 = 5/6, so 5π/6.
- Write the angle over 180, for example 135/180.
- Reduce the fraction by the greatest common divisor. 135 and 180 share 45, so 135/180 = 3/4.
- Put π in the numerator: 3π/4.
It works for awkward angles too: 20° = 20/180 = π/9, 50° = 5π/18 and 1° = π/180. Only switch to decimals (0.872665 rad for 50°) at the very end, when you need a number for a calculator or a program.
Sources and further reading
The standards and references behind the numbers on this page. Links open in a new tab.
Standards and official sources
Degrees to radians FAQs
Short, exact answers to what people ask most.
How do you convert 45 degrees to radians?
Multiply 45 by π/180. That gives 45π/180, which reduces to π/4, or about 0.785398 rad.
What is 1 degree in radians?
One degree is π/180 radians, about 0.01745329 rad. There are 360 degrees and 2π radians in a full circle.
What is 180 degrees in radians?
180 degrees is exactly π radians, about 3.141593 rad. That is the basis of every degree to radian conversion.
How many radians are in 360 degrees?
360 degrees equals 2π radians, about 6.283185 rad. It is one full turn of a circle.
Why does my calculator give the wrong sine of 30?
It is probably in RAD mode, so it treats 30 as 30 radians and returns −0.988. Switch to DEG mode, or convert 30° to π/6 rad first, and you get 0.5.
How do I convert degrees to radians in Excel?
Use the RADIANS function, for example =RADIANS(30), or multiply by PI()/180. Excel's SIN, COS and TAN all expect radians.
What is 90 degrees in radians?
90 degrees is π/2 radians, about 1.570796 rad, a right angle.
What is 120 degrees in radians?
120 degrees is 2π/3 radians, about 2.094395 rad.
How many radians is 10 degrees?
10 degrees is π/18 radians, about 0.174533 rad.
What is 15 degrees in radians?
15 degrees is π/12 radians, about 0.261799 rad, half of π/6.
Related conversions
Definitions from the BIPM SI Brochure (9th edition), which defines the radian and lists the degree as a non-SI unit accepted for use with the SI, and from ISO 80000-3 for plane angle; the mean Earth radius of 6,371 km from the IUGG. Last updated . How we verify our numbers.