Degrees to Radians Converter (and Back)

To convert degrees to radians, multiply the angle by π/180 and simplify the fraction. For example, 135° × π/180 = 135π/180 = 3π/4, which is about 2.3562 radians. To go back from radians to degrees, multiply by 180/π: 5π/6 × 180/π = 150°. One radian is about 57.2958°.

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How to use the degrees to radians converter

1. Choose a direction: Degrees → radians, Radians → degrees, DMS → radians for angles written in degrees, minutes and seconds, or Any angle unit for gradians, turns, arcminutes, arcseconds and milliradians.
2. Type the angle. Degrees can be whole numbers, decimals, fractions or negative. Radians can use π, typed as π or pi, such as 3pi/4.
3. Read the exact answer in terms of π and the decimal. Below it you get the same angle in every other unit, a unit circle picture, the coterminal angle, quadrant and reference angle.
4. Open Show steps to see the multiplication and how the fraction was simplified. Change the decimal places at the top if your teacher wants a different rounding.

Degrees to radians formula

A full turn is 360°, and it is also 2π radians, because the circumference of a circle is 2π times its radius. Halve both and you get the key fact: 180° = π radians. Everything else follows from it.

• Degrees to radians: radians = degrees × π/180
• Radians to degrees: degrees = radians × 180/π
• 1° = π/180 ≈ 0.017453 radians, and 1 radian = 180/π ≈ 57.2958°

A radian is the angle that cuts off an arc exactly as long as the radius. That is why radians fit naturally into arc length (s = rθ), sector area and calculus, while degrees are easier for everyday angles.

Worked examples

Example 1: 60° to radians

Multiply by π/180: 60 × π/180 = 60π/180. The greatest common factor of 60 and 180 is 60, so 60π/180 = π/3. As a decimal, π/3 ≈ 1.0472 radians.

Example 2: 210° to radians in terms of π

210 × π/180 = 210π/180. Divide the top and bottom by 30 to get 7π/6. Check: 7π/6 is a little more than π, and 210° is a little more than 180°, so the answer is sensible.

Example 3: 5π/6 and 2 radians to degrees

5π/6 × 180/π: the π signs cancel, leaving 5 × 180 ÷ 6 = 150°. For 2 radians there is no π to cancel: 2 × 180/π = 360/π ≈ 114.5916°, or 114° 35′ 29.61″.

Example 4: a DMS angle to radians

The latitude 40° 26′ 46″ becomes 40 + 26/60 + 46/3600 ≈ 40.4461° in decimal degrees. Then 40.4461 × π/180 ≈ 0.7059 radians.

Common angles in degrees and radians

A quick memory trick: learn 30° = π/6, 45° = π/4 and 60° = π/3. Every other angle in the table is a whole-number multiple of one of them, so 150° = 5 × 30° = 5π/6.

Degrees, minutes and seconds (DMS)

Navigation, astronomy and surveying split a degree into 60 minutes (′) and a minute into 60 seconds (″). To use a DMS angle in a radian formula, first make it decimal degrees with D + M/60 + S/3600, then multiply by π/180. Going the other way, the whole part is the degrees, 60 times the decimal part gives minutes, and 60 times what is left gives seconds. A negative sign, or S and W on a map, applies to the whole angle.

Gradians and revolutions

Some calculators have a GRAD mode. A gradian is 1/400 of a turn, so a right angle is 100 grad and 1 grad = 0.9°. A revolution (turn) is one full circle: 360°, 2π radians or 400 grad. If a calculator answer looks strange, check whether it is set to DEG, RAD or GRAD.

Common mistakes when converting angles

Checking your answer and doing it by hand

Use these quick checks before you hand in work: 180° should become π, so any angle under 180° gives fewer than π ≈ 3.14 radians. Radians are always about 57 times smaller than the degree number. Multiplying your answer back by 180/π should return the original angle. On homework, work the first problem or two by hand, then use the converter to check the rest and to see where a mistake crept in.

Where radians are used

Radians are the standard unit in trigonometry beyond right triangles, in graphs of sine and cosine, in calculus (the derivative of sin x is cos x only when x is in radians), and in physics for angular speed. Degrees stay common in geometry, construction, maps and everyday directions. The unit circle calculator shows sine, cosine and tangent for any of these angles, and the reference angle calculator walks through reference and coterminal angles in detail.

Degrees and radians practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you convert degrees to radians in terms of π?

Multiply the degrees by π/180 and keep π as a symbol. Then simplify the number in front of π. For 135°, 135/180 reduces to 3/4, so the answer is 3π/4. For 210°, 210/180 reduces to 7/6, so the answer is 7π/6.

How many radians are in 1 degree?

One degree is π/180 radians, about 0.017453. So 10° is π/18 ≈ 0.1745 radians. In the other direction, 1 radian is 180/π, about 57.2958°.

Is 180 degrees equal to π radians?

Yes. Half a turn is 180°, and it is also π radians, because a full turn is 2π radians. This single fact gives both conversion factors, π/180 and 180/π.

How do you convert radians to degrees without π?

Multiply by 180 and divide by π ≈ 3.14159. For 2 radians, 2 × 180 ÷ 3.14159 ≈ 114.59°. The answer is not a whole number because there is no π to cancel.

Why do mathematicians prefer radians?

Radians measure an angle by arc length, so formulas such as s = rθ, A = ½r²θ and the derivatives of sine and cosine need no extra conversion factor. Degrees are an arbitrary split of the circle into 360 parts.

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• Unit circle calculator
• Reference angle calculator
• Arc length and sector area calculator
• Trigonometry calculator
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