Calculus, physics and programming work in radians while most geometry uses degrees. Convert either way and see the result as a multiple of π, the form most textbooks use.
Enter the angle.
Choose the direction of conversion.
Read the converted value.
Degrees to radians: rad = deg × π ÷ 180
Radians to degrees: deg = rad × 180 ÷ π
Angles can be measured in degrees, where a full turn is 360°, or in radians, where a full turn is 2π. This converter changes a value from degrees to radians or from radians to degrees, and also shows the result as a multiple of π, the form textbooks use, such as 45° = 0.25π = π/4.
Degrees are familiar from protractors and Class 7 geometry, but calculus, physics and programming all work in radians. Formulas such as arc length s = rθ, angular speed v = ωr and the derivative of sin x being cos x are true only when angles are in radians. Programming languages including Python, JavaScript and C expect radians in their sin and cos functions, which catches many beginners. A quick conversion avoids a whole class of errors.
1. To convert degrees to radians, multiply by π and divide by 180: radians = degrees × π ÷ 180.
2. To express the answer as a multiple of π, divide the degrees by 180 and simplify the fraction, for example 150 ÷ 180 = 5/6, so 150° = 5π/6.
3. To convert radians to degrees, multiply by 180 and divide by π: degrees = radians × 180 ÷ π.
4. If a radian value is given as a multiple of π, replace π by 180°, for example 3π/4 = 3 × 180° ÷ 4 = 135°.
5. Use 1 radian ≈ 57.2958° and 1° ≈ 0.0174533 radians for quick estimates.
6. Check that a full turn gives 360° = 2π and a right angle gives 90° = π/2.
Take a circle of radius r and mark off an arc along its edge that is also r long. The angle this arc makes at the centre is one radian. Since the whole circumference is 2πr, it contains 2π such arcs, so a full turn is 2π radians. Equating this with 360° gives π radians = 180°, and every conversion follows from that one fact. A radian is a ratio of two lengths, arc over radius, so it has no physical unit.
In radians, arc length is simply s = rθ and sector area is ½r²θ, with no conversion factors. For small angles, sin θ ≈ θ, which physicists use for pendulums and optics. Most importantly, the derivative of sin x is cos x only in radians; in degrees it would be (π/180) cos x. Degrees come from ancient Babylonian astronomy and are convenient because 360 has many divisors, but radians are the natural unit that makes calculus clean.
Some scientific calculators also offer gradians, where a right angle is 100 grad and a full turn is 400 grad, used in some surveying. Degrees can be split into minutes and seconds: 1° = 60′ and 1′ = 60″, so 30°15′ is 30.25°. Convert these to decimal degrees before using this converter. Common exact values worth memorising are 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2 and 180° = π.
Aman is writing a Python program to draw a clock face and needs the angle of 150° in radians, because math.sin expects radians.
Radians = degrees × π ÷ 180: 150 × 3.14159265 ÷ 180 = 2.617994 rad = 0.833333 π
Answer: Radians 2.617994; As a multiple of π 0.833333 π
Multiplying by 180 ÷ π when converting degrees to radians, which is the reverse conversion.
Passing degree values straight into programming sin and cos functions, which assume radians.
Treating the π in 5π/6 as a variable instead of about 3.14159.
Entering degrees-minutes-seconds such as 30°15′ as 30.15 instead of 30.25.
Leaving a calculator in the wrong DEG or RAD mode after converting.
Preparing angles for sin, cos and tan functions in Python, JavaScript, Excel and C.
Solving calculus and physics problems on arcs, sectors and circular motion.
Converting angular speed between degrees per second and radians per second.
Checking standard angle values like π/6 and π/4 in trigonometry.
Working with rotation angles in CAD, robotics and game engines.
How many degrees is 1 radian?
About 57.2958 degrees.
Why do calculators have a DEG/RAD mode?
Because sin, cos and tan give different results depending on whether the angle is read as degrees or radians.