What is the formula to convert radians to degrees?

Multiply the radian value by 180, then divide by pi (approximately 3.14159265). Written as a formula: degrees = radians times 180 divided by pi. For example, pi/2 rad times 180 divided by pi equals 90 degrees.

How many degrees is one radian?

One radian equals 180 divided by pi, which is approximately 57.29578 degrees. This is because a full circle spans 2 pi radians and 360 degrees, so each radian covers 360 divided by (2 pi) degrees.

How do you convert degrees to radians?

Multiply the degree value by pi, then divide by 180. For example, 90 degrees times pi divided by 180 equals pi/2 radians (approximately 1.5707963 rad). This calculator also supports this direction — just switch the dropdown to "Degrees to Radians".

What does the degrees-minutes-seconds (DMS) output mean?

DMS notation splits the fractional part of a degree into arc-minutes (1/60 of a degree) and arc-seconds (1/3600 of a degree). For instance, 143.2394 degrees becomes 143 degrees, 14 minutes, and 21.84 seconds. This format is used in surveying, GPS coordinates, and astronomy.

Why do trig functions use radians rather than degrees?

In calculus, the derivative of sin(x) equals cos(x) only when x is measured in radians. If degrees were used, a correction factor of pi/180 would appear in every derivative. Radians also make arc-length calculations natural: arc length equals radius times angle in radians, with no conversion factor needed.

What is the unit-circle point shown by the calculator?

Any angle theta placed on the unit circle (a circle of radius 1 centred at the origin) corresponds to a point (cos theta, sin theta). This is how cosine and sine are defined geometrically. The calculator displays this point so you can immediately see where your angle lands on the circle.

Radians to Degrees Calculator

This radians-to-degrees calculator converts any angle between the two most common angle units, shows the step-by-step arithmetic, and goes further than a plain converter: it also computes all six trigonometric functions, expresses the angle in degrees-minutes-seconds (DMS) notation, identifies the quadrant, gives the pi-fraction equivalent where it exists, and reports the corresponding unit-circle coordinates. Whether you are a student checking homework, an engineer working in robotics, or a developer debugging a graphics shader, every value you need is on one page with no network requests.

How it works

The relationship between degrees and radians comes from the circumference of a circle. A full turn of 360 degrees is the same angle as one full revolution in radians, which covers an arc length equal to the circumference 2 pi r. On a unit circle (r = 1) the arc length of a full turn equals 2 pi, so:

360 degrees = 2 pi radians

Dividing both sides by 2 gives:

180 degrees = pi radians

Rearranging for the two conversion directions:

degrees = radians times (180 / pi)
radians = degrees times (pi / 180)

The constant 180 / pi is approximately 57.29578, so multiplying any radian value by that number gives degrees. The constant pi / 180 is approximately 0.01745329, so multiplying any degree value by that gives radians.

This calculator uses JavaScript’s Math.PI constant (a 64-bit IEEE 754 double with 15-16 significant digits) and Math.sin, Math.cos, and Math.tan for trig values, so results are as precise as the platform allows.

Worked example

Convert 2.5 radians to degrees:

degrees = 2.5 times (180 / pi)
degrees = 2.5 times 57.29578
degrees = 143.2394°

Convert back to verify: 143.2394 times (pi / 180) = 143.2394 times 0.01745329 = 2.5000 radians. Round-trip confirmed.

DMS breakdown of 143.2394°:

Whole degrees: 143°
Fractional degree: 0.2394 times 60 = 14.364 → 14′ (arc-minutes)
Fractional minute: 0.364 times 60 = 21.84″ (arc-seconds)
Result: 143° 14′ 21.84″

Trig values at 2.5 rad:

Function	Value
sin(2.5 rad)	0.598472
cos(2.5 rad)	−0.801144
tan(2.5 rad)	−0.747022
cot(2.5 rad)	−1.338645
sec(2.5 rad)	−1.248217
csc(2.5 rad)	1.670530

Unit-circle coordinates: (cos 2.5, sin 2.5) = (−0.801144, 0.598472). The angle sits in quadrant II because the normalised angle (143.2°) is between 90° and 180°.

Formula note

The core identity pi radians = 180 degrees is exact and dimensionless — it is a ratio of arc length to radius. All other conversions follow by proportion. Trig functions like sin and cos are defined as ratios of sides in a right triangle or, equivalently, as the y- and x-coordinates of a point on the unit circle at angle theta. These definitions are valid for any real angle, positive or negative, greater than one revolution or less than zero, using the radian measure directly with no adjustment.

The six trig functions shown follow from two primary ones:

cot(theta) = cos(theta) / sin(theta) — reciprocal direction of tan
sec(theta) = 1 / cos(theta) — reciprocal of cosine
csc(theta) = 1 / sin(theta) — reciprocal of sine

When cos(theta) = 0 (at 90°, 270°, and so on) then sec and tan are undefined; when sin(theta) = 0 (at 0°, 180°) then csc and cot are undefined. The calculator shows “undefined” for these cases rather than a misleading large number.