How to Convert Degrees to Radians
Converting degrees to radians is one of the most important operations in mathematics, physics, and engineering. Degrees are the familiar unit of angular measure used in everyday life, navigation, and surveying, while radians are the standard unit of angular measure in higher mathematics, calculus, and physics. The radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle, making it a natural and dimensionless measure of angle. This conversion is essential for anyone using trigonometric functions in calculus, since derivatives and integrals of sine and cosine are only correct when the angle is measured in radians. Programmers working with graphics, game development, and robotics constantly convert between these units because mathematical libraries typically expect radian input. Physicists describing rotational motion, angular velocity, and oscillatory phenomena also require radians. Surveyors, architects, and GPS systems often work in degrees but must convert to radians for computational geometry. Mastering this conversion is foundational for success in STEM disciplines.
Conversion Formula
The formula converts degrees to radians by multiplying the degree value by the factor pi/180. This factor arises because a full circle is 360 degrees or 2pi radians. Dividing both sides by 360 gives 1 degree = 2pi/360 = pi/180 radians. Therefore, to convert any angle from degrees to radians, multiply the number of degrees by pi/180, which is approximately 0.0174533. This relationship is fundamental to the definition of radian measure.
rad = deg × π/180
5 degrees = 0.0872665 radians
Step-by-Step Example
To convert 5 degrees to radians:
1. Start with the value: 5 degrees
2. Multiply by the conversion factor: 5 × π/180
3. Calculate: 5 × 0.0174533 = 0.0872665
4. Result: 5° = 0.0872665 radians
Understanding Degrees and Radians
What is a Degree?
The degree as a unit of angular measure dates back to ancient Babylon, around 3000 BCE. The Babylonians used a base-60 (sexagesimal) number system, and their astronomers divided the circle into 360 parts, possibly because 360 approximates the number of days in a year and is highly divisible. This convention was transmitted through Greek mathematics, notably through Ptolemy's Almagest, and became the global standard for angular measurement in navigation, cartography, astronomy, and everyday use. The degree remains the most widely recognized unit of angle in practical applications.
What is a Radian?
The concept of radian measure emerged in the 18th century as mathematicians sought a more natural unit for angular measurement in analysis. Roger Cotes first described the radian concept in 1714, though he did not use the term. The word "radian" was coined by James Thomson (brother of Lord Kelvin) in 1873. The radian was adopted as the SI supplementary unit for plane angle in 1960 and was later reclassified as a derived SI unit equal to one meter per meter (m/m), emphasizing its dimensionless nature. Today, the radian is the standard angular unit in all branches of pure and applied mathematics.
Practical Applications
This conversion is used extensively in computer programming, where trigonometric functions in most languages (C, Python, JavaScript, Java) expect arguments in radians. Game developers converting character rotation angles from user-friendly degree inputs to radian values for rendering engines use this formula constantly. In physics, angular velocity is measured in radians per second, so converting degree-based rotation data to radians is a routine step. Robotics engineers programming joint angles and aerospace engineers calculating satellite orbital parameters both rely on this conversion. In electrical engineering, phase angles of alternating current signals are frequently converted from degrees to radians for phasor analysis and Fourier transforms.
Tips and Common Mistakes
A very common error is dividing by pi/180 instead of multiplying, which effectively converts radians to degrees rather than the other way around. Another frequent mistake is using 360 instead of 180 in the formula, which yields a result that is off by a factor of two. When working with programming languages, be aware that some functions like atan2 return radians while you might expect degrees; always check the documentation. A useful reference: 90° = pi/2, 180° = pi, and 360° = 2pi. If your result for 90° is not approximately 1.5708, recheck your formula.
Frequently Asked Questions
Multiply the degree value by π/180. For example, 45° becomes 45 × π/180 = π/4 ≈ 0.7854 radians. This works because a full circle of 360° equals 2π radians, making each degree equal to π/180 radians.