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Radians to Degrees Converter

Convert radians to degrees with our free radians to degrees converter — or flip it and turn degrees back into radians. A half turn of π radians equals 180 degrees, so one radian is about 57.2958 degrees, and this page converts either way instantly.

Common angles

How it works

Degrees and radians are two ways of measuring the same thing: the size of an angle. Degrees split a full circle into 360 equal parts, a system inherited from ancient Babylonian astronomy, and it is the system most people meet first at school and in everyday life. Radians measure an angle by the circle itself. One radian is the angle you get when the arc of the circle is exactly as long as the radius, which makes radians the natural unit for mathematics, physics, and engineering.

The bridge between the two units is the number π. A full circle is 360 degrees, and it is also 2π radians, because the circumference of a circle is 2π times its radius. Half a circle is therefore 180 degrees, or π radians. That single fact gives both conversion formulas:

To convert radians to degrees, multiply by 180 and divide by π (about 57.2958 degrees per radian). To convert degrees to radians, multiply by π and divide by 180 (about 0.0174533 radians per degree). The converter above does both directions: type the angle, pick the direction, and use the swap button to flip it.

It helps to learn the common angles by heart, because they appear constantly in trigonometry. A right angle of 90 degrees is π/2 radians, about 1.5708. A 45-degree angle is π/4 radians, about 0.7854. A 30-degree angle is π/6 radians, about 0.5236, and a 60-degree angle is π/3 radians, about 1.0472. Notice how the radian values are neat fractions of π — that is no accident, and it is one reason mathematicians prefer radians: the angles that matter most have the simplest values.

Radians also make formulas simpler. In radians, the length of an arc is just the radius multiplied by the angle, with no conversion factor at all. The derivatives of sine and cosine are clean in radians, and the small-angle approximations used in physics only work when the angle is measured in radians. That is why calculators have a radian mode, why programming languages expect angles in radians, and why physics textbooks switch to radians as soon as rotation, waves, or circular motion appear.

One practical trap is mixing the two units up. If a calculator is set to degrees while a formula expects radians, the answer can be wildly wrong — a sine of 1 radian is 0.841, while a sine of 1 degree is only 0.017. Whenever a result looks strange, the first thing to check is the angle mode. This converter is a quick way to translate between the two before plugging a number into a formula.

Worked example

A student is working on a trigonometry problem and needs to convert π radians — 3.14159 — into degrees.

Step 1: Write down the angle — 3.14159 radians.
Step 2: Use the formula — degrees = radians × 180 ÷ π.
Step 3: Substitute — 3.14159 × 180 ÷ 3.14159 = 180.
Step 4: Round to two decimal places — 180.00°.

That is the exact result this converter gives for π radians: 180 degrees, a half turn. Going the other way, 90 degrees converts to radians by multiplying 90 × π ÷ 180, which gives 1.5708 radians — a quarter turn, or π/2. Both directions use the same π bridge, just in reverse.

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