How to Convert Degrees to Radians (and Radians to Degrees)

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 150° × π/180 = 5π/6 radians ≈ 2.618, and 2.5 radians × 180/π ≈ 143.24°. One full turn is 360°, which equals 2π radians, so 180° = π radians is the only fact you need to remember.

The conversion formulas

Both come from the same equivalence, 180° = π rad. Dividing it through gives the two key constants:

• 1° = π/180 ≈ 0.0174533 rad
• 1 rad = 180/π ≈ 57.2958°

Degrees to radians, step by step

1. Multiply the degree measure by π/180.
2. Simplify the fraction if you want an exact answer in terms of π.
3. Multiply out with π ≈ 3.14159 if you want a decimal.

Simplifying is just reducing the fraction degrees/180. Since 150/180 = 5/6, the answer is 5π/6.

Radians to degrees, step by step

If the angle contains π, replace π with 180° and simplify:

If it is a plain decimal, multiply by 180/π:

The angle converter handles degrees, radians, gradians, turns and arcminutes in either direction.

Common angles

A quick pattern: multiples of 30° are multiples of π/6, and multiples of 45° are multiples of π/4. These are the angles on the unit circle with exact sine and cosine values; the unit circle calculator shows them all, and the trig functions of π calculator evaluates expressions like sin(5π/6) exactly.

What a radian actually is

Place the radius of a circle along its edge, bending it to follow the curve. The angle at the center that this arc subtends is one radian. Because the circumference is 2πr, exactly 2π radius-lengths fit around the circle, which is why a full turn is 2π radians.

That definition makes radians a ratio of two lengths, so they are technically dimensionless. The International System of Units (SI) still names the radian as the derived unit for plane angle, and NIST’s guide to the SI follows that convention.

Why radians make formulas simpler

Arc length and sector area

With θ in radians:

In degrees, the same formulas need an extra factor of π/180 every time. The circle sector calculator accepts either unit, and how to find the area of a circle covers whole circles.

Angular speed

Rotational speed in physics and engineering is measured in radians per second. A motor turning at 3,000 revolutions per minute spins at 3,000 × 2π ÷ 60 ≈ 314.16 rad/s. Multiplying angular speed by radius gives the linear speed at the edge, which only works directly in radians.

Calculus and small angles

The derivative of sin x is cos x only when x is measured in radians. Radians also give the small-angle approximation: for small x, sin x ≈ x. For instance, sin(0.1) = 0.09983, very close to 0.1. In degrees, no such neat relationship exists.

Degrees, minutes and seconds

Navigation and mapping often split degrees into minutes (′) and seconds (″), with 60 minutes per degree and 60 seconds per minute.

Convert DMS to decimal degrees first, then to radians. Historically, one minute of latitude along Earth’s surface defined the nautical mile, now fixed at exactly 1,852 meters. The decimal degrees to DMS converter does both directions.

Other angle units

Gradians appear on some surveying instruments and on the GRAD setting of scientific calculators, which is a common source of accidental errors.

Estimating in your head

Think of every angle as a fraction of 180°, then attach π. 60° is one-third of 180°, so it is π/3. 20° is one-ninth of 180°, so it is π/9 ≈ 0.35 rad. For a quick decimal, use the rule that 1 radian is a little under 60°: 3 radians is about 172°, just short of a half turn (π ≈ 3.14 rad = 180°). Going the other way, 1° is about 0.0175 rad, so 10° ≈ 0.175 rad and 40° ≈ 0.70 rad (exact: 0.698).

Negative angles and angles beyond 360°

The formulas work unchanged for any angle. −90° is −π/2, and 720°, two full turns, is 4π. Angles that differ by full turns (multiples of 360° or 2π) point in the same direction and are called coterminal; 405° and 45° are coterminal, as are 9π/4 and π/4. Trig functions give identical values for coterminal angles, so you can subtract 2π repeatedly to bring any radian measure into the range 0 to 2π before looking it up.

Common mistakes

• Using 180/π when you mean π/180. Converting degrees to radians always makes the number much smaller, since you multiply by about 0.0175. If 90° turns into 5,157, you multiplied by 180/π instead.
• Treating “π” as optional. 5π/6 and 5/6 are very different angles: about 150° versus 47.7°.
• Rounding π early. Keep exact forms through a calculation and convert to a decimal only at the end.

Calculator mode matters

Every scientific calculator has a DEG/RAD setting. Entering sin(30) in radian mode returns about −0.988 instead of 0.5, because it computes the sine of 30 radians. Spreadsheets and programming languages always use radians: in Excel, =SIN(30) returns −0.988, so write =SIN(RADIANS(30)) instead. Excel’s RADIANS() and DEGREES() functions do the conversions in this guide.

To find the equivalent angle between 0° and 360° for large or negative angles, use the reference angle calculator. For the right-triangle ratios themselves, see SOHCAHTOA explained.