Degree-Radian Conversion
Simple Explanation
A radian is defined as the angle created when an arc's length exactly equals the circle's radius. Since a full circle's circumference is 2Οr, a full revolution is 2Ο radians β the same 360Β°. This gives the conversion: radians = degrees Γ (Ο/180).
Why Do We Need It?
Radians are the natural unit for angles in higher mathematics and physics (calculus formulas and circular motion equations assume radians) β fluently converting between degrees and radians is essential going forward.
See It
A circle showing an arc whose length equals the radius, with the corresponding central angle marked as 1 radian, approximately 57.3 degrees
Formula
Degree-Radian Conversion
radians = degrees Γ (Ο/180) degrees = radians Γ (180/Ο)
Converts an angle measurement between degrees and radians, the two standard units for measuring angles.
- Ο
- β pi, approximately 3.14159 β one full revolution is 2Ο radians, or 360Β°
When to use it: Whenever an angle is given in one unit but a formula or calculation requires the other.
Worked Example
Convert degrees to radians
Convert 150Β° to radians (leave your answer in terms of Ο).
Why Does This Work?
A full revolution measures 360Β° and also measures 2Ο radians (since the full circumference, 2Οr, contains exactly 2Ο "radius-length" arcs) β setting 360Β° = 2Ο radians and simplifying gives 1Β° = Ο/180 radians, the exact conversion factor.
Real-Life Example
Programming a robot arm's rotation
Many robotics and graphics programming libraries require rotation angles to be entered in radians, not degrees.
An engineer converting a design spec (given in degrees) to code must apply exactly this conversion formula before the robot arm will rotate correctly.
Practice
Convert Ο/3 radians to degrees.
MediumCommon mistake
Multiplying by 180/Ο when converting degrees to radians (or Ο/180 when converting radians to degrees) β these are inverses of each other, and using the wrong one gives a wildly incorrect result.
Quick Review
- radians = degrees Γ (Ο/180); degrees = radians Γ (180/Ο).
- 360Β° = 2Ο radians; 180Β° = Ο radians.
- A radian is the angle where arc length equals the radius.