Degree to Radian Converter
Enter the value that you want to convert degree (°) to radian (rad) or radian to degree.Also written as ° to RAD conversion.
1 degree = 0.0174533 radian
Formula: radian = degree value × 0.0174533
° to RAD — degree to radian
10 degree = 0.17453 radian
Popular Degree Converters:
Degrees to Radians in Programming & Math
Converting degrees to radians is one of the most common operations in programming. JavaScript's Math.sin(), Math.cos(), and Math.atan2() all expect radians, but user interfaces typically display degrees. Game engines like Unity use degrees in the inspector but radians internally. CSS transforms accept degrees, while WebGL shaders require radians. The formula is simple — multiply by π/180 — but having a quick reference table prevents bugs in rotation code, physics simulations, and trigonometric calculations.
Degree vs Radian
What is Degree?
Dating back to ancient Babylonian astronomy, the degree divides a full rotation into 360 equal parts.
The Babylonians used a base-60 number system and observed that the sun moved approximately one degree per day across the sky over a year. The 360° circle became universal in navigation, surveying, architecture, and education. Degrees remain the standard angle unit in everyday life — from compass bearings and map coordinates to smartphone screen rotation and camera field-of-view specifications.
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.
degree to radian metric conversion table
| 0.01 ° | = | 0.00017 rad |
| 0.1 ° | = | 0.00175 rad |
| 1 ° | = | 0.01745 rad |
| 2 ° | = | 0.03491 rad |
| 3 ° | = | 0.05236 rad |
| 4 ° | = | 0.06981 rad |
| 5 ° | = | 0.08727 rad |
| 6 ° | = | 0.10472 rad |
| 7 ° | = | 0.12217 rad |
| 8 ° | = | 0.13963 rad |
| 9 ° | = | 0.15708 rad |
| 10 ° | = | 0.17453 rad |
| 11 ° | = | 0.19199 rad |
| 12 ° | = | 0.20944 rad |
| 13 ° | = | 0.22689 rad |
| 14 ° | = | 0.24435 rad |
| 15 ° | = | 0.26180 rad |
| 16 ° | = | 0.27925 rad |
| 17 ° | = | 0.29671 rad |
| 18 ° | = | 0.31416 rad |
| 19 ° | = | 0.33161 rad |
| 20 ° | = | 0.34907 rad |
| 30 ° | = | 0.52360 rad |
| 40 ° | = | 0.69813 rad |
| 50 ° | = | 0.87266 rad |
| 60 ° | = | 1.04720 rad |
| 70 ° | = | 1.22173 rad |
| 80 ° | = | 1.39626 rad |
| 90 ° | = | 1.57080 rad |
| 100 ° | = | 1.74533 rad |
| 200 ° | = | 3.49066 rad |
| 300 ° | = | 5.23599 rad |
| 400 ° | = | 6.98132 rad |
| 500 ° | = | 8.72665 rad |
| 600 ° | = | 10.47198 rad |
| 700 ° | = | 12.21730 rad |
| 800 ° | = | 13.96263 rad |
| 900 ° | = | 15.70796 rad |
| 1000 ° | = | 17.45329 rad |
How to Convert ° to RAD (Degree to Radian)?
We can convert degree to radian by using an example.
Example:
Convert 20 Degree to Radian?
We know 1 Degree = 0.0174533 radian; 1 Radian = 57.2957795 degree.
20 degree = ___rad
20 × 0.0174533 = 0.34907 rad (we know 1 degree = 0.0174533 radian)
Answer:
20 degree = 0.34907 radian
How to Convert Degrees to Radians (Step-by-Step)
The conversion from degrees to radians uses the relationship that 180° equals π radians. Follow these steps to convert any angle from degrees to radians:
- Write down the angle value in degrees that you want to convert.
- Multiply the degree value by π/180 (approximately 0.0174533).
- Simplify the resulting fraction if possible (e.g., 90 × π/180 = π/2).
- Express the final answer in terms of π for exact values, or as a decimal for approximate values.
Common Degrees to Radians Conversions
The following reference table shows standard angle conversions that appear frequently in mathematics, physics, and engineering. Memorizing these common values significantly speeds up trigonometry calculations.
| Degree | Radian |
|---|---|
| 0° | 0 rad |
| 30° | π/6 ≈ 0.5236 rad |
| 45° | π/4 ≈ 0.7854 rad |
| 60° | π/3 ≈ 1.0472 rad |
| 90° | π/2 ≈ 1.5708 rad |
| 120° | 2π/3 ≈ 2.0944 rad |
| 135° | 3π/4 ≈ 2.3562 rad |
| 150° | 5π/6 ≈ 2.6180 rad |
| 180° | π ≈ 3.1416 rad |
| 210° | 7π/6 ≈ 3.6652 rad |
| 270° | 3π/2 ≈ 4.7124 rad |
| 360° | 2π ≈ 6.2832 rad |
Solved Examples
Question 1: Convert 200 degrees into radians.
Solution:
Angle in radians = Angle in degrees × π/180
= 200 × π/180
= 200π/180
= 10π/9 (simplify by dividing both by 20)
Answer: 200° = 10π/9 radians ≈ 3.4907 rad
Question 2: Convert 450 degrees into radians.
Solution:
Angle in radians = Angle in degrees × π/180
= 450 × π/180
= 450π/180
= 5π/2 (simplify by dividing both by 90)
Answer: 450° = 5π/2 radians ≈ 7.8540 rad
Question 3: Convert 75 degrees to radians.
Solution:
Angle in radians = 75 × π/180
= 75π/180
= 5π/12 (simplify by dividing both by 15)
Answer: 75° = 5π/12 radians ≈ 1.3090 rad
Question 4: Convert −120 degrees to radians.
Solution:
The negative sign indicates clockwise rotation.
Angle in radians = −120 × π/180
= −120π/180
= −2π/3 (simplify by dividing both by 60)
Answer: −120° = −2π/3 radians ≈ −2.0944 rad
Question 5: Write 22.5° in radians as a fraction of π.
Solution:
Angle in radians = 22.5 × π/180
= 22.5π/180
= π/8 (since 22.5/180 = 1/8)
Answer: 22.5° = π/8 radians ≈ 0.3927 rad
Practice Questions
Try solving these on your own to test your understanding:
- Convert 700 degrees to radians. (Answer: 35π/9 ≈ 12.217 rad)
- How many radians is equal to 100 degrees? (Answer: 5π/9 ≈ 1.7453 rad)
- Convert 550 degrees to radians. (Answer: 55π/18 ≈ 9.599 rad)
- Express 15° in radians as a fraction of π. (Answer: π/12)
- Convert 330° to radians. (Answer: 11π/6 ≈ 5.7596 rad)
- What is −45° in radians? (Answer: −π/4 ≈ −0.7854 rad)
Converting Negative Degrees to Radians
Negative angles represent clockwise rotation. The conversion method is identical — multiply by π/180. For example: −90° × (π/180) = −π/2 radians. Negative angles are common in coordinate geometry, navigation (bearing below the reference line), and physics (clockwise torque). The sign simply indicates direction of rotation, not magnitude.
Why π/180 is the Conversion Factor
A full circle contains 360° or 2π radians. Dividing both by 360 gives: 1° = 2π/360 = π/180 radians ≈ 0.0174533 rad. This relationship exists because radians are defined by the arc-length-to-radius ratio of a circle. When the arc length equals the radius, the angle is 1 radian (≈ 57.296°). Since the full circumference is 2πr, a full circle is 2π radians.
Degrees to Radians in Programming Languages
Most programming languages provide trigonometric functions that expect radians. Here are quick conversion patterns: JavaScript: radians = degrees * (Math.PI / 180). Python: import math; radians = math.radians(degrees). C/C++: radians = degrees * M_PI / 180.0. In game development, Unity uses Mathf.Deg2Rad as a constant multiplier, while Unreal Engine's FMath::DegreesToRadians() handles the conversion internally.
Key Takeaways
- Multiply degrees by π/180 (≈ 0.0174533) to get radians.
- 180° = π radians is the fundamental relationship.
- Common angles: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2.
- Negative degrees convert the same way — the sign is preserved.
- Most programming math libraries require angles in radians, not degrees.
- 1 radian ≈ 57.296° — this is useful for quick mental estimation.
Degree to Radian Conversion Formula
radian = degree × 0.0174533
1 degree = 0.0174533 radian
1 radian = 57.2957795 degree
Reverse: degree = radian × 57.2957795
Frequently Asked Questions
How many radian are in 1 degree?
There are 0.0174533 radian in 1 degree. To convert degree to radian, multiply the value by 0.0174533.
How do I convert degree to radian?
Multiply your degree value by 0.0174533 to get the equivalent in radian. For example, 5 degree = 5 × 0.0174533 = 0.08727 radian.
How do I convert radian to degree?
Multiply your radian value by 57.2957795 to get the equivalent in degree. Alternatively, divide by 0.0174533.
What is 10 degree in radian?
10 degree is equal to 0.17453 radian.
What is 100 degree in radian?
100 degree is equal to 1.74533 radian.