Turn to Radian Converter
Enter the value that you want to convert turn (tur) to radian (rad) or radian to turn.
1 turn = 6.2831853 radian
Formula: radian = turn value × 6.2831853
TUR to RAD — turn to radian
10 turn = 62.83185 radian
Popular Degree Converters:
Converting Turn to Radian
Understanding the Turn
The turn (also called revolution or full rotation) represents one complete 360° cycle.
Turns are used in engineering to count rotational cycles — RPM (revolutions per minute) is effectively turns per minute. They appear in motor specifications, propeller speeds, centrifuge settings, and vinyl record playback speeds. In mathematics, the τ (tau) constant equals one turn (2π radians).
The Radian: History & Usage
The radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius.
Adopted as the SI unit of angular measure, the radian makes calculus and physics equations significantly simpler because it eliminates conversion constants. One full revolution equals 2π radians (≈ 6.2832 rad). Radians are essential in signal processing, robotics, computer graphics (3D rotations), oscillation analysis, and any domain where angular velocity or frequency calculations are performed.
Why Angle Conversions Matter
Angle conversions bridge the gap between everyday measurement (degrees) and mathematical computation (radians). A mechanical engineer designing a robotic arm needs radians for inverse kinematics. A pilot reads compass headings in degrees. A surveyor might work in gradians. Game developers convert between degrees (artist-friendly) and radians (GPU shaders) hundreds of times per frame. Understanding these conversions is essential across STEM fields, navigation, and digital media production.
turn to radian metric conversion table
| 0.01 tur | = | 0.06283 rad |
| 0.1 tur | = | 0.62832 rad |
| 1 tur | = | 6.28319 rad |
| 2 tur | = | 12.56637 rad |
| 3 tur | = | 18.84956 rad |
| 4 tur | = | 25.13274 rad |
| 5 tur | = | 31.41593 rad |
| 6 tur | = | 37.69911 rad |
| 7 tur | = | 43.98230 rad |
| 8 tur | = | 50.26548 rad |
| 9 tur | = | 56.54867 rad |
| 10 tur | = | 62.83185 rad |
| 11 tur | = | 69.11504 rad |
| 12 tur | = | 75.39822 rad |
| 13 tur | = | 81.68141 rad |
| 14 tur | = | 87.96459 rad |
| 15 tur | = | 94.24778 rad |
| 16 tur | = | 100.53096 rad |
| 17 tur | = | 106.81415 rad |
| 18 tur | = | 113.09734 rad |
| 19 tur | = | 119.38052 rad |
| 20 tur | = | 125.66371 rad |
| 30 tur | = | 188.49556 rad |
| 40 tur | = | 251.32741 rad |
| 50 tur | = | 314.15927 rad |
| 60 tur | = | 376.99112 rad |
| 70 tur | = | 439.82297 rad |
| 80 tur | = | 502.65482 rad |
| 90 tur | = | 565.48668 rad |
| 100 tur | = | 628.31853 rad |
| 200 tur | = | 1256.63706 rad |
| 300 tur | = | 1884.95559 rad |
| 400 tur | = | 2513.27412 rad |
| 500 tur | = | 3141.59265 rad |
| 600 tur | = | 3769.91118 rad |
| 700 tur | = | 4398.22972 rad |
| 800 tur | = | 5026.54825 rad |
| 900 tur | = | 5654.86678 rad |
| 1000 tur | = | 6283.18531 rad |
How to Convert TUR to RAD (Turn to Radian)?
We can convert turn to radian by using an example.
Example:
Convert 20 Turn to Radian?
We know 1 Turn = 6.2831853 radian; 1 Radian = 0.1591549 turn.
20 turn = ___rad
20 × 6.2831853 = 125.66371 rad (we know 1 turn = 6.2831853 radian)
Answer:
20 turn = 125.66371 radian
How to Convert Turns to Radians
One full turn equals 2π radians. This conversion is fundamental in physics and engineering:
- Write down the number of turns.
- Multiply by 2π (approximately 6.2832).
- The result is the angle in radians.
Solved Examples
Question 1: A motor runs at 50 turns per second. What is its angular velocity in rad/s?
Solution:
Angular velocity ω = turns/s × 2π
= 50 × 2π
= 100π
≈ 314.16 rad/s
Answer: 50 rev/s = 100π ≈ 314.16 rad/s
Question 2: Convert 0.25 turns (quarter turn) to radians.
Solution:
Radians = 0.25 × 2π
= π/2
≈ 1.5708
Answer: 1/4 turn = π/2 radians
Question 3: A wheel makes 1200 RPM. Convert to rad/s.
Solution:
First convert RPM to turns/second: 1200/60 = 20 turns/s
Then: ω = 20 × 2π = 40π ≈ 125.66 rad/s
Answer: 1200 RPM = 40π ≈ 125.66 rad/s
Practice Questions
Try solving these on your own to test your understanding:
- Convert 3 turns to radians. (Answer: 6π ≈ 18.85 rad)
- A centrifuge at 10000 RPM — what is ω in rad/s? (Answer: ≈ 1047.2 rad/s)
- Convert 1/6 turn to radians. (Answer: π/3 ≈ 1.047 rad)
- Angular displacement of 7.5 turns in radians? (Answer: 15π ≈ 47.12 rad)
- A fan blade at 900 RPM — angular velocity in rad/s? (Answer: 30π ≈ 94.25 rad/s)
The ω = 2πf Formula
The most common use of turns-to-radians conversion in physics is the angular frequency formula: ω = 2πf, where f is frequency in Hz (cycles per second = turns per second) and ω is angular frequency in rad/s. This appears everywhere: AC electricity (50/60 Hz → 314/377 rad/s), sound waves, orbital mechanics, and quantum mechanics. Every frequency expressed in "hertz" is implicitly in turns/second.
Moment of Inertia and Rotational Energy
Rotational kinetic energy KE = ½Iω² requires ω in rad/s, not RPM. When given a flywheel spinning at 5000 RPM, you must convert: ω = 5000 × 2π/60 = 523.6 rad/s before calculating energy. Similarly, torque τ = Iα requires angular acceleration α in rad/s². This is why the turns-to-radians conversion is essential in every rotational dynamics problem.
Key Takeaways
- Multiply turns by 2π (≈ 6.2832) to get radians.
- ω (rad/s) = RPM × 2π/60 = RPM × 0.10472.
- Angular frequency ω = 2πf (where f is in Hz = turns/s).
- Always convert to radians before using in physics formulas.
- Rotational energy KE = ½Iω² requires ω in rad/s, not RPM.
Turn to Radian Conversion Formula
radian = turn × 6.2831853
1 turn = 6.2831853 radian
1 radian = 0.1591549 turn
Reverse: turn = radian × 0.1591549
Frequently Asked Questions
How many radian are in 1 turn?
There are 6.2831853 radian in 1 turn. To convert turn to radian, multiply the value by 6.2831853.
How do I convert turn to radian?
Multiply your turn value by 6.2831853 to get the equivalent in radian. For example, 5 turn = 5 × 6.2831853 = 31.41593 radian.
How do I convert radian to turn?
Multiply your radian value by 0.1591549 to get the equivalent in turn. Alternatively, divide by 6.2831853.
What is 10 turn in radian?
10 turn is equal to 62.83185 radian.
What is 100 turn in radian?
100 turn is equal to 628.31853 radian.