How does a degrees-to-radians converter work?

Any converter follows the two relationships above. The steps are straightforward:

1. Determine your starting unit. Are you entering degrees or radians? This is important because the formula changes direction depending on where you’re coming from.
2. Apply the correct conversion.
   
   • If you start with degrees, multiply by π and divide by 180 to get radians.
     radians=degrees×(π/180).
   • If you start with radians, multiply by 180 and divide by π to get degrees.
     degrees=radians×(180/π).
3. Round to your preferred precision. Most converters let you choose how many decimal places to display. For everyday use, two or three decimals are plenty; for engineering or programming, you might choose six or more.

Behind the scenes, the converter’s job is simply to perform these multiplications or divisions and handle the digits for you. Some also show the radian result as a multiple of π—for example, they might report 0.785398 radians as π/4, which is the exact fraction.

What do degrees measure?

Degrees break a full circle into 360 equal parts. That’s why right angles are 90° (a quarter turn), straight lines are 180° (a half turn) and a full rotation is 360°. We use degrees in everyday life because they’re intuitive: a 45° slice is easy to imagine as half of a right angle.

What about radians?

Radians come from geometry. Instead of counting out 360 pieces, radians measure how far you travel around a circle compared to its radius. If you walk along the edge of a circle and travel a distance exactly equal to the radius, the angle you’ve “turned through” is one radian. Because a full circle’s circumference is 2 π times the radius, there are 2 π radians in 360°. That means π radians equal 180°. Rearranging this gives the two key conversion facts you’ll use:

• 1 radian equals 180 divided by π degrees, which works out to about 57.2958°.
• 1 degree equals π divided by 180 radians (because if 180° is π radians, dividing both sides by 180 gives you 1°).

With these two relationships, you can move back and forth between degrees and radians easily.

Why do people bother with radians?

Radians can look odd at first because they involve π, but they simplify a lot of math. For example, when you use radians, the circumference formula C=2πrC = 2\pi rC=2πr and the arc length formula s=rθs = r\thetas=rθ (with θ in radians) come together nicely. Calculus also works more naturally with radians—derivatives of sine and cosine functions behave cleanly only when angles are in radians. Programmers often need to supply angles in radians because many computer graphics libraries expect them.

Example: converting degrees to radians

Let’s say you’re working on a DIY project and need to cut a piece of wood at a 60° angle. You’re using a program that measures angles in radians. Here’s how the conversion happens:

1. Start with 60°.
2. Multiply by π and divide by 180:
   60°×(π/180)=(60/180)​π=(3/1)​π.
3. Written as a decimal, that’s roughly 1.0472 radians.
   

So you’d enter approximately 1.0472 radians into your software or note that it’s exactly π/3.

Example: converting radians to degrees

Now suppose you’re working with an electronics library that gives you an angle of 1.5 radians. You want to visualize this in degrees to better understand it. Using the other formula:

1. Start with 1.5 radians.
2. Multiply by 180 and divide by π:
   1.5×(180/π)≈1.5×57.2958≈85.9437°.
3. So 1.5 radians is about 85.94°, just shy of a right angle.

Seeing the degree value helps you picture how far the rotation goes.

Tips for working with degrees and radians

• Remember key benchmarks. If you memorize that 90° = π/2, 180° = π and 360° = 2π, you can quickly estimate other conversions.
• Choose your format. When you see something like 0.5236 radians, you can write it as π/6 for an exact fraction. Conversely, writing 45° as π/4 is more compact in algebra.
• Pay attention to units in software. Many programming languages use radians by default for trigonometric functions. Forgetting to convert degrees can lead to confusing errors.
• Use a converter for accuracy. While it’s good to understand the math, letting a tool handle π and decimal precision reduces mistakes.