Degrees to Radians Converter

Degrees vs Radians: What’s the Difference?

Degrees and radians are two ways to measure angles. Degrees are based on dividing a circle into 360 equal parts, which makes them intuitive for everyday geometry and navigation. Radians are based on the radius of a circle, which makes them the natural unit for trigonometry, calculus, and many physics formulas.

A key relationship connects them: one full turn is 360°, and it is also 2π radians. That’s why π appears in every conversion. This converter lets you move between the two systems quickly while also showing the exact π-based fraction when possible.

How to Convert Degrees to Radians

The formula is simple: radians = degrees × (π/180). You can think of it as “scale degrees down” because 180 degrees maps to π, which is about 3.14159. In other words, degrees count slices of a circle, while radians measure the arc length of a unit circle.

What if you want the reverse conversion? Use degrees = radians × (180/π). Many calculators and programming libraries assume radians by default, which is why this conversion is so common in STEM work.

Why Radians Matter in Trigonometry and Calculus

In pure math, radians simplify many formulas. For example, derivatives of sin(x) and cos(x) take their cleanest form when x is measured in radians. In physics, angular velocity, oscillations, and wave equations are also typically written using radians.

In practice, this means a common “why is my answer wrong?” issue is angle mode. If you enter 30 expecting 30° but your calculator is using radians, sin(30) will not equal 0.5. Converting to the correct unit (or switching mode) removes that confusion.

Exact π Form vs Decimal Radians

Some degree values convert neatly into a fraction of π. For example, 45° becomes π/4 and 90° becomes π/2. This is called “exact π form” and it’s useful for proofs, trig identities, and learning. Other values don’t simplify nicely, so a decimal approximation is more practical.

This tool shows both forms: the decimal (useful for computation and programming) and the π form (useful for learning and exact symbolic reasoning). You can also adjust precision depending on whether you need a quick estimate or more digits.

Common Angles to Remember

Many trig problems use a small set of angles repeatedly. It’s worth memorizing a few:

• 30° = π/6
• 45° = π/4
• 60° = π/3
• 90° = π/2
• 180° = π
• 360° = 2π

Having these in mind helps you sanity-check results instantly. If you convert 180° and you don’t see π (or about 3.14159), something is off in the input or formatting.

Negative Angles and Angles Larger Than 360°

Angles aren’t limited to a single rotation. Negative angles represent rotation in the opposite direction, and angles larger than 360° represent multiple turns. The conversion formula doesn’t change—you can convert any real degree value to radians by multiplying by π/180.

If you’re comparing angles or simplifying them, you might also reduce them “mod 360°” (or “mod 2π” in radians), but that’s a separate step from converting units.

Degrees to Radians in Programming

In most programming languages and math libraries, trig functions expect radians. That means if you have a degree value from a user interface or a sensor output, you typically convert it before calling sin, cos, or tan. A common pattern is: rad = deg * Math.PI / 180.

This converter is a quick way to verify the expected radian input before you paste values into code or validate results from a calculation.

Limitations and Practical Notes

The π form shown here is simplified to common fractions when possible. For arbitrary degree values, a perfect “nice fraction” may not exist, so the decimal representation is the best practical output. Always choose a precision that fits your use case: more digits for computation, fewer digits for readability.