Radians to Degrees Converter
Free Radians degrees Calculator for angles. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Formula
Degrees = Radians x (180 / pi)
To convert radians to degrees, multiply by 180/pi (approximately 57.2958). To convert degrees to radians, multiply by pi/180 (approximately 0.01745). This relationship comes from the fact that a full circle is both 360 degrees and 2*pi radians.
Worked Examples
Example 1: Converting Pi/4 Radians to Degrees
Problem: Convert pi/4 radians to degrees, DMS notation, and find the trig values.
Solution: Degrees = (pi/4) x (180/pi) = 45 degrees\nDMS = 45 deg 0' 0\"\nsin(pi/4) = sqrt(2)/2 = 0.7071\ncos(pi/4) = sqrt(2)/2 = 0.7071\ntan(pi/4) = 1.0000\nQuadrant: I
Result: pi/4 radians = 45 degrees (Quadrant I)
Example 2: Converting 2.5 Radians to Degrees
Problem: Convert 2.5 radians to degrees and identify the quadrant.
Solution: Degrees = 2.5 x (180/pi) = 2.5 x 57.2958 = 143.239 degrees\nDMS = 143 deg 14' 20.14\"\nNormalized: 143.239 degrees\nQuadrant: II (between 90 and 180)\nsin(2.5) = 0.5985, cos(2.5) = -0.8011
Result: 2.5 radians = 143.239 degrees (Quadrant II)
Frequently Asked Questions
What is the relationship between radians and degrees?
Radians and degrees are two different units for measuring angles. One complete revolution around a circle equals 360 degrees or 2*pi radians (approximately 6.2832 radians). Therefore, 1 radian equals 180/pi degrees (approximately 57.2958 degrees), and 1 degree equals pi/180 radians (approximately 0.01745 radians). The conversion formulas are: degrees = radians times (180/pi) and radians = degrees times (pi/180). Radians are dimensionless because they represent the ratio of arc length to radius, making them the natural unit for angle measurement in mathematics and physics.
Why do mathematicians prefer radians over degrees?
Mathematicians prefer radians because they simplify many formulas and make calculus cleaner. When angles are in radians, the derivative of sin(x) is simply cos(x), and the derivative of cos(x) is -sin(x). In degrees, these derivatives require an extra factor of pi/180. The Taylor series for sin(x) = x - x^3/3! + x^5/5! - ... only works when x is in radians. Arc length on a circle is simply s = r*theta when theta is in radians. The small angle approximations sin(x) approximately equals x and tan(x) approximately equals x only hold in radians. Euler's formula e^(ix) = cos(x) + i*sin(x) requires radians. Essentially, radians eliminate conversion factors from nearly every trigonometric and calculus formula.
How do you convert DMS (degrees, minutes, seconds) to radians?
To convert DMS notation to radians, first convert everything to decimal degrees, then multiply by pi/180. For example, convert 45 degrees 30 minutes 15 seconds: the minutes component equals 30/60 = 0.5 degrees, the seconds component equals 15/3600 = 0.004167 degrees. Total decimal degrees = 45 + 0.5 + 0.004167 = 45.504167 degrees. Multiply by pi/180: 45.504167 times 0.017453 = 0.79418 radians. Going the reverse direction: convert radians to decimal degrees (multiply by 180/pi), then extract degrees as the integer part, multiply the fractional part by 60 for minutes, and multiply the fractional minutes by 60 for seconds.
How are radians used in physics and engineering?
Radians are the standard angle unit in physics and engineering equations. Angular velocity is measured in radians per second (rad/s), not degrees per second. The relationship v = r*omega (linear velocity equals radius times angular velocity) requires omega in radians. Torque equations, moment of inertia calculations, and rotational dynamics all use radians. In electrical engineering, AC voltage is described by V(t) = V0*sin(omega*t), where omega (angular frequency) is in radians per second. Phase differences between signals are expressed in radians. Structural engineers use radians when calculating beam deflections and structural analysis involving trigonometric functions.
What are gradians and how do they compare to radians and degrees?
Gradians (also called gons or grads) are a third angle measurement system where a right angle equals 100 gradians and a full circle equals 400 gradians. One gradian equals 0.9 degrees or pi/200 radians. Gradians were introduced during the French Revolution as part of the metric system and are still used in some European countries for surveying and civil engineering. The conversion formulas are: gradians = degrees times (10/9) and gradians = radians times (200/pi). While gradians make right angles and percentages of a turn easy to express (a right angle is exactly 100 grad), they never gained the mathematical elegance of radians or the widespread familiarity of degrees.
Can radians be negative or greater than 2pi?
Yes, radian measures can be any real number, positive, negative, or beyond 2pi. Positive angles are measured counterclockwise from the positive x-axis, while negative angles go clockwise. So -pi/2 radians is 90 degrees clockwise, equivalent to 270 degrees or 3pi/2 radians counterclockwise. Angles greater than 2pi represent multiple revolutions: 3pi radians is 1.5 revolutions (540 degrees), and 4pi radians is 2 complete revolutions (720 degrees). Coterminal angles differ by multiples of 2pi and share the same trigonometric values. In physics, cumulative angles matter: a wheel rotating 10pi radians has made 5 complete turns, even though its angular position is equivalent to 0 radians.