Radians to Degrees Converter

radians to degrees — quick reference

The formula, worked line by line

The radian-to-degree factor is derived rather than memorised. Begin with π rad = 180° and divide both sides by π. The result is 1 rad = 180/π degrees, so multiplying by 180/π replaces radians with degrees in one step.

For a plain numerical radian input, leave π in the denominator until the final evaluation. The exact expression for 1 rad is 180/π degrees. Because π is irrational, its decimal expansion does not terminate, and the converter rounds the result to two decimal places.

If the radian input is itself a multiple of π, simplify before evaluating. The π in π/2 rad cancels the π in the conversion factor, leaving 90° exactly. This cancellation explains why familiar circle angles remain tidy while values such as 1 rad do not.

π rad = 180°
1 rad = 180/π degrees
degrees = radians × 180 ÷ π
1 rad → 1 × 180 ÷ π ≈ 57.30°

The three π placement mistakes

“Is radians to degrees 180 pi?” No. The operation is multiplication by 180/π, not 180π. The defining equation already contains π beside the radian value, so isolating one radian requires dividing both sides by π rather than multiplying the degree side by it.

“Is 1 radian 180 degrees?” No. π radians is 180°, while 1 rad is approximately 57.2958°. Replacing π rad with 1 rad discards the factor π and turns a half-turn statement into a false claim about a much smaller angle.

“Is 360 degrees 1 radian?” No. Since 180° equals π rad, doubling both quantities gives 360° = 2π rad. A complete turn is therefore approximately 6.2832 radians, while one radian covers only about 57.30° of that turn.

How to use the radians to degrees converter

Enter a radian value and read the degree equivalent to two decimal places. The default 1 rad becomes approximately 57.30°. That number is not arbitrary: one radian is the central angle that cuts off an arc exactly as long as the circle’s radius.

Start with π rad = 180°. Divide both sides by π and the conversion factor appears: 1 rad = 180/π degrees. Multiply the input by 180/π. This single derivation is safer than trying to recall whether π belongs above, below or beside 180.

Plain radian numbers usually produce untidy degree answers. A rational input such as 1 or 2 is multiplied by 180/π, and π remains in the denominator of the exact expression. The decimal is therefore useful for interpretation, but it can never be an exact finite representation.

Multiples of π are the important exception. When the radian input contains π, that π can cancel against the denominator in 180/π. Thus π/6 rad becomes 30°, π/4 rad becomes 45°, and π/2 rad becomes 90° exactly. The neatness comes from cancellation, not coincidence.

Use the degree result to picture an angle or communicate with an audience accustomed to degrees. Keep radians inside calculus, physics and code when the governing formula expects them. Converting units for presentation is helpful; changing units halfway through a derivation without adjusting the formula is not.

When the angle belongs to a larger geometry problem, the Circle Calculator relates central angles to arcs and sectors, and the Triangle Calculator relates angles to sides. Those tools continue the calculation after this page has made the unit readable.

What one radian actually measures

Draw a circle with radius r, then mark an arc whose length is also r. The central angle facing that arc is one radian. Because angle in radians is arc length divided by radius, r ÷ r = 1, whatever length unit was used for both measurements.

That construction explains why one radian is approximately 57.29577951308232° rather than a memorable whole-degree angle. The circumference contains 2π radii, so one complete turn contains 2π radians. Splitting the same turn into 360 equal degree parts creates a different numerical scale.

Radians are dimensionless in SI because the defining quantities are two lengths in a ratio. The name “radian” remains essential in communication, however, because it tells the reader that the dimensionless number represents a plane angle rather than an unrelated ratio with the same numerical value.

This is also why radians are natural in calculus. When θ is in radians, arc length is simply radius × θ, and the small-angle approximation sin θ ≈ θ has its direct form. Degrees would insert an additional π/180 factor into those relationships.

Why plain radians give untidy degrees

The two directions are not visually symmetric. A rational degree number gains a factor of π and usually becomes a clean rational multiple of π radians. A nonzero rational radian number instead gains 180/π, leaving irrationality in the denominator and usually producing a non-terminating degree decimal.

For 2 rad, the exact expression is 360/π degrees and the two-place display is approximately 114.59°. There is nothing unfinished about that decimal-looking answer. The input simply was not a familiar fraction of π, so no π cancels to reveal a whole degree number.

A decimal input that merely approximates a π multiple deserves special care. For example, 1.5708 rad is close to π/2 but is not exactly π/2. It converts to approximately 90.00° at two displayed places, yet the rounded display must not be used to claim exact equality.

Every radians value, worked out

33 common radians figures each get their own page, with the answer at full precision, the arithmetic, what rounding costs, and the nearest real-world reference point on the scale.

Questions people ask