degrees to radians
Unit Converters
degreesradians
r
=
d
×
(
π
180
)
- r (Radians): The angular measurement in radians.
- d (Degrees): The angular measurement in degrees.
- π (Pi): Approximately 3.14159.
Degree to Radian Conversion Table
| Degree [°] | Radian [rad] |
|---|---|
| 0.01 ° | 0.0001745329 rad |
| 0.1 ° | 0.0017453293 rad |
| 1 ° | 0.0174532925 rad |
| 2 ° | 0.034906585 rad |
| 3 ° | 0.0523598776 rad |
| 5 ° | 0.0872664626 rad |
| 10 ° | 0.1745329252 rad |
| 20 ° | 0.3490658504 rad |
| 50 ° | 0.872664626 rad |
| 100 ° | 1.745329252 rad |
| 1,000 ° | 17.4532925199 rad |
What is degrees to radians?
Converting between degrees and radians is a fundamental shift in how we perceive the geometry of a circle. While degrees are the traditional unit for navigation and construction, radians are the "natural" unit used in calculus, physics, and higher mathematics. Understanding the conversion between them requires looking at the very definition of a circle's circumference.
Understanding the Values
Degree (°): A degree is a measurement of a plane angle in which one full rotation is 360 equal parts. It is an arbitrary unit based on historical counting systems rather than a physical property of the circle itself.
Radian (rad): A radian is the "SI" unit for angular measurement. It is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. This makes it a "dimensionless" unit because it represents a ratio of two lengths.
π (Pi): The mathematical constant representing the ratio of a circle's circumference to its diameter (approximately 3.14159). In the conversion formula, π acts as the bridge between the linear distance of the arc and the angular sweep.
History and Origin
The history of these units reveals the evolution of human thought from practical observation to abstract reasoning.
The Babylonian Degrees
The 360-degree system is ancient, dating back to the Babylonians. They used a sexagesimal (base-60) number system. It is believed they chose 360 because it is close to the 365 days in a solar year and is highly composite—meaning it can be easily divided by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, and 24. This made it incredibly practical for early astronomers and sailors who needed to divide the sky into manageable sections without complex fractions.
The Rise of the Radian
The concept of the radian was not formally named until the 1870s, though mathematicians had been using the underlying principle for centuries. The term was coined by James Thomson (brother of Lord Kelvin). Unlike degrees, which were "invented" by humans, radians were "discovered" as an inherent property of circles.
In the 17th century, as calculus began to emerge with the work of Newton and Leibniz, mathematicians realized that trigonometric functions (like sin and cos) behave much more simply when the input is in radians. If you use degrees in calculus, messy conversion factors (180/π) appear in every derivative and integral. By using radians, the math becomes "clean," which is why radians are the undisputed standard in modern science and engineering.
What is Degree?
Definition & Usage
A degree (symbol: °) is a measurement of a plane angle in which one full rotation is 360 degrees.
What is Radian?
Definition & Usage
The radian (symbol: rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.
Frequently Asked Questions
How accurate is this degrees to radians tool?
Our tools utilize high-precision floating point math guaranteeing accuracy up to the 6th decimal place.
Is this free to use?
Yes, all converters and calculators on ToolsMetrics are 100% free with no limits.