Understanding Angle Measurement: Degrees and Radians
Angles are fundamental in mathematics, physics, engineering, and many other fields. They describe rotation, the spread between two lines, or the arc of a circle. While you might be most familiar with measuring angles in "degrees," another crucial unit of angular measurement is the "radian." This calculator helps you effortlessly convert between these two systems.
What are Degrees?
A degree (symbol: °) is a unit of plane angle, defined such that a full circle is 360 degrees. This system dates back to ancient Babylonians, who used a sexagesimal (base-60) number system. The choice of 360 is convenient because it is highly composite, meaning it has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360), making it easy to divide a circle into many common fractions.
- A right angle is 90°.
- A straight angle is 180°.
- A full rotation is 360°.
What are Radians?
A radian (symbol: rad) is the standard unit of angular measure in many areas of mathematics, particularly calculus and physics. It is defined based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
The circumference of a circle is given by 2πr. Since an arc of length r subtends one radian, a full circle (arc length 2πr) subtends 2π radians. This means:
- A full rotation is
2πradians. - A straight angle is
πradians. - A right angle is
π/2radians.
The beauty of radians lies in their natural connection to the geometry of a circle and their simplification of many mathematical formulas, especially in calculus (e.g., the derivative of sin(x) is cos(x) only when x is in radians).
Why Convert Between Degrees and Radians?
Different contexts and tools often require angles in specific units:
- Everyday Use: Degrees are more intuitive for general communication and navigation.
- Mathematics & Science: Radians are preferred in higher-level mathematics, physics, and engineering because they simplify formulas and make calculations more elegant, especially when dealing with trigonometric functions, rotational motion, and wave phenomena.
- Programming: Many programming languages' trigonometric functions (like
Math.sin(),Math.cos()in JavaScript) expect angles in radians.
How to Use the Angle Conversion Calculator
Our calculator provides two distinct conversion tools:
- Radians to Degrees: Enter an angle in radians into the first input field and click "Convert to Degrees." The result will show the equivalent angle in degrees.
- Degrees to Radians: Enter an angle in degrees into the second input field and click "Convert to Radians." The result will show the equivalent angle in radians.
You can use the "Clear" button to reset both input fields and the result.
Conversion Formulas:
- Radians to Degrees:
Degrees = Radians × (180 / π) - Degrees to Radians:
Radians = Degrees × (π / 180)
Examples:
- Example 1: Convert
π/2radians to degrees.(π/2) × (180 / π) = 90° - Example 2: Convert
180°to radians.180 × (π / 180) = π radians - Example 3: Convert
1radian to degrees.1 × (180 / π) ≈ 57.2958° - Example 4: Convert
45°to radians.45 × (π / 180) = π/4 radians ≈ 0.7854 radians
Use the calculator above to quickly perform these and other angle conversions!