How to Convert Radians to Degrees

One radian is about 57.3 degrees. That's the whole conversion in a nutshell, yet it trips people up constantly, especially when a function like Math.atan2() returns an angle such as 2.356 and you need to show it to a user as 135°.

Converting radians to degrees takes one multiplication. Here's the formula, the shortcuts, and the historical reason we measure circles in 360 parts at all.

The Number 360 Probably Came From Astronomy, Not Geometry

There's nothing mathematically special about dividing a circle into 360 parts. The choice is usually traced to ancient Babylonian astronomers, who used a base-60 number system and tracked the Sun's roughly year-long journey through the sky. A year of about 360 days maps neatly onto a circle, and 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many more numbers, which makes fractions of a circle easy.

Radians, by contrast, come from the circle's own geometry. Degrees are the human convention; radians are the natural unit.

The Formula

Degrees = Radians × 180 / π

Because a full circle is 2π radians = 360°, half a circle gives:

π radians = 180°
1 radian = 180/π ≈ 57.2958°

Convert instantly with the radians to degrees calculator.

Worked Examples

Example 1: An Angle With π

When the angle includes π, the π's cancel:

π/3 × 180/π = 180/3 = 60°
5π/4 × 180/π = 5 × 180/4 = 225°

Shortcut: replace π with 180 and simplify.

Example 2: A Plain Decimal

2 radians × 180/π ≈ 2 × 57.2958 ≈ 114.59°

Example 3: The atan2 Result

2.356 × 180/π ≈ 134.99° ≈ 135°

Conversion Table

Angles Larger Than a Full Turn

Rotations can exceed 2π. To find the equivalent angle between 0° and 360°, convert and then subtract multiples of 360:

7.5 radians × 180/π ≈ 429.72°
429.72° − 360° = 69.72°

For negative angles, add 360° until the result is positive: −1 radian ≈ −57.30°, which is equivalent to 302.70°. The reference angle calculator does this automatically.

Converting to Degrees, Minutes and Seconds

Sometimes you need the result in DMS form, as in navigation or astronomy.

Example: 1 radian ≈ 57.2958°

1. Whole degrees: 57°
2. Multiply the decimal part by 60: 0.2958 × 60 ≈ 17.75 → 17′
3. Multiply the remaining decimal by 60: 0.75 × 60 ≈ 45 → 45″

So 1 radian ≈ 57° 17′ 45″. Check with the DMS converter.

An Insider Reference: Ptolemy's Minutes and Seconds

Around 150 AD, the Greek astronomer Claudius Ptolemy of Alexandria wrote the Almagest, the most influential astronomy text for over a thousand years. It included a table of chords, an early form of trigonometry, measured in degrees and subdivided into sixtieths, following Babylonian base-60 practice.

Medieval Latin translations called the first division pars minuta prima, "the first small part," and the second pars minuta secunda, "the second small part." Those became our words minute and second, for both angles and time.

Not every system stuck with 360. During the French Revolution, reformers introduced the gradian, dividing a right angle into 100 parts and a full circle into 400. It still appears on many scientific calculators as "GRAD." Explore all three with the angle unit converter.

Where You'll Need This Conversion

• Programming: functions like atan2, acos and asin return radians. Convert before displaying an angle to users, such as a compass heading or a joystick direction.
• Physics: angular velocity is usually given in radians per second. A wheel turning at 10 rad/s spins about 573° every second, or roughly 1.6 full turns.
• Engineering drawings and surveying: plans and field instruments typically use degrees, while the calculations behind them often use radians.
• Spreadsheets: Excel and Google Sheets provide a DEGREES() function that performs exactly this conversion.

Common Mistakes

1. Dividing instead of multiplying. Radians → degrees multiplies by 180/π. If a small radian value becomes even smaller, you've used the wrong factor.

2. Leaving π unconverted. An answer like "60π°" is wrong. The π's should cancel.

3. Using the wrong calculator mode. Check DEG vs. RAD before trusting any trig result.

Two Concepts Worth Knowing

Radian

A radian is the angle whose arc length equals the radius. There are 2π radians in a full circle.

Coterminal Angles

Coterminal angles end at the same position, like 30°, 390° and −330°. They differ by whole multiples of 360° (or 2π radians). See the trigonometric formulas.

Quick Answer: How Do You Convert Radians to Degrees?

Multiply the angle in radians by 180/π. For example, π/3 radians × 180/π = 60°, and 2 radians × 180/π ≈ 114.59°. One radian is about 57.2958°. If the result is more than 360°, subtract 360° to find the equivalent angle.

Try Them Yourself

• Radians to Degrees Calculator: convert any angle
• Degrees to Radians Calculator: the reverse
• DMS Converter: degrees, minutes and seconds
• Reference Angle Calculator: normalize large or negative angles
• Angle Unit Converter: degrees, radians and gradians
• How to Convert Degrees to Radians: why radians matter

Convert 1, 2, 3 and 6 radians to degrees without a calculator, using 1 radian ≈ 57.3°. Then check how close 6 radians comes to a full circle, and explain why it falls just short.