Degrees to Radians Converter
Our free angles calculator solves degrees radians problems. Get worked examples, visual aids, and downloadable results.
Formula
radians = degrees * (pi/180), degrees = radians * (180/pi)
Since a full circle is both 360 degrees and 2*pi radians, the conversion factor is pi/180 (about 0.01745) for degrees to radians, and 180/pi (about 57.296) for radians to degrees. One radian equals the angle where the arc length equals the radius.
Worked Examples
Example 1: Converting 135 Degrees to Radians
Problem: Convert 135 degrees to radians and express as a fraction of pi.
Solution: 135 * (pi/180) = 135*pi/180\nSimplify: GCD(135, 180) = 45\n135/45 = 3, 180/45 = 4\nResult: 3*pi/4 = 3 * 3.14159 / 4 = 2.3562 radians\nQuadrant: II (between 90 and 180)\nReference angle: 180 - 135 = 45 degrees = pi/4
Result: 135 degrees = 3*pi/4 = 2.3562 radians (Quadrant II, reference angle 45 degrees)
Example 2: Converting 2.5 Radians to Degrees
Problem: Convert 2.5 radians to degrees and find the quadrant.
Solution: 2.5 * (180/pi) = 2.5 * 57.2958 = 143.239 degrees\nQuadrant: II (between 90 and 180)\nReference angle: 180 - 143.239 = 36.761 degrees\nsin(2.5) = 0.5985, cos(2.5) = -0.8011\nBoth signs confirm Quadrant II (sin positive, cos negative).
Result: 2.5 radians = 143.239 degrees (Quadrant II). sin = 0.599, cos = -0.801.
Frequently Asked Questions
How do you convert degrees to radians?
To convert degrees to radians, multiply the degree value by pi/180. This conversion factor comes from the fact that a full circle is both 360 degrees and 2*pi radians, so the ratio is 2*pi/360 = pi/180. For example, 90 degrees = 90 * (pi/180) = pi/2 radians. Another way to think about it: divide the degree measure by 180, then multiply by pi. Common conversions worth memorizing include 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, and 180 degrees = pi radians. These relationships form the foundation for working with trigonometric functions in mathematics and science.
Why do mathematicians prefer radians over degrees?
Radians are preferred in mathematics because they create the simplest possible formulas in calculus and analysis. The derivative of sin(x) equals cos(x) only when x is in radians. If x were in degrees, the derivative would include an extra factor of pi/180. Similarly, the fundamental limit lim(sin(x)/x) = 1 as x approaches 0 only holds in radians. The Taylor series for trigonometric functions also require radians: sin(x) = x - x^3/6 + x^5/120 - ... only works for radians. Arc length formula s = r*theta is cleanest in radians with no conversion factor. The Euler formula e^(i*theta) = cos(theta) + i*sin(theta) also requires radians. These mathematical simplifications make radians the natural choice for analysis.
What are gradians and how do they relate to degrees and radians?
Gradians (also called grads or gons) divide a right angle into 100 parts, making a full circle 400 gradians. This metric-inspired system was introduced during the French Revolution as part of the metrification effort. Conversions: 1 degree = 10/9 gradians, 1 gradian = 0.9 degrees = pi/200 radians. Gradians are primarily used in surveying and some European engineering applications because 100 gradians equals a right angle, making percentage-of-slope calculations easier. However, gradians never achieved widespread adoption in mathematics or physics, where radians dominate. Scientific calculators typically offer three angle modes: degrees (DEG), radians (RAD), and gradians (GRAD), so always verify which mode is active before computing.
What is a turn and how does it compare to degrees and radians?
A turn (also called a revolution or cycle) is the simplest angular unit: one turn equals a full rotation. Half a turn is a semicircle, quarter turn is a right angle. Conversions: 1 turn = 360 degrees = 2*pi radians = 400 gradians. Turns are intuitive for describing rotational motion: a wheel making 3.5 turns is easier to visualize than 1260 degrees or 7*pi radians. In some programming languages and graphics frameworks (notably LOGO), turns are the primary angle unit. The tau movement (tau = 2*pi = 6.2832) advocates using tau instead of pi because 1 tau equals exactly 1 turn, making many formulas more intuitive. For example, a quarter circle is tau/4 rather than pi/2.
Can I use Degrees to Radians Converter on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Can I share or bookmark my calculation?
You can bookmark the calculator page in your browser. Many calculators also display a shareable result summary you can copy. The page URL stays the same so returning to it will bring you back to the same tool.