Degrees to Radians: How to Convert

This calculator converts an angle given in degrees to its equivalent in radians, the standard unit of angular measure used in mathematics and physics. You enter the angle in degrees, and you get the radian value, often expressed as a multiple of π for exactness. No prior knowledge is needed beyond basic arithmetic.

Formula

The conversion relies on the fundamental relationship: a full circle is 360° and also 2π radians. Therefore, to convert degrees to radians, multiply the number of degrees by π/180.

Radians = Degrees × (π / 180)

• Degrees – the angle you start with, measured in degrees (°).
• Radians – the resulting angle, measured in radians (rad).
• π – the mathematical constant, approximately 3.14159.

Worked Example

Using the default value of 180 degrees:

1. Write down the formula: Radians = Degrees × (π / 180).
2. Substitute 180 for degrees: Radians = 180 × (π / 180).
3. Simplify: 180 divided by 180 equals 1, so Radians = 1 × π = π.
4. The exact result is π radians, and the decimal approximation is 3.141593 (rounded to six decimal places).

This matches the calculator’s output: the exact answer shows π (as 3.1416 when rounded to four decimals) and the decimal answer shows 3.141593.

When to Use This Conversion

Radians are the standard unit for angles in calculus, trigonometry, and most scientific formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. When you are working with equations like angular velocity (ω = θ/t) or arc length (s = rθ), the angle must be in radians. Converting degrees to radians is also essential when using programming functions like Math.sin() in JavaScript or math.sin() in Python, which expect radians.

Common Mistakes

• Forgetting to multiply by π/180 – Some people simply use the degree number as if it were already radians. Always apply the conversion factor.
• Using 180/π instead of π/180 – That would convert radians to degrees, not the other way around. Double-check the fraction orientation.
• Rounding π too early – If you round π to 3.14 before multiplying, you lose accuracy. Keep π as a symbol or use a full precision value until the final step.
• Mixing up degrees and grads – Some calculators have a “grad” mode (400 grads per circle). Make sure your calculator is in degree mode when entering the original angle.

Tips

• For exact answers, leave the result as a multiple of π (e.g., 45° = π/4). This is often preferred in theoretical work.
• For practical applications, use a decimal approximation with enough digits. Six decimal places are usually sufficient.
• Memorize common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π. These appear frequently.
• When using a calculator, check the mode setting (DEG vs. RAD) before performing trigonometric calculations.

Troubleshooting

Your calculator gives a different result than expected. Check if the calculator is in degree mode when you enter the original angle. If it is in radian mode, it will interpret the number as radians, not degrees.

The result seems too small or too large. A full circle is 2π ≈ 6.283 radians. If your result for a full circle is not about 6.283, you likely used the wrong conversion factor.

You are not sure whether to use exact π or a decimal. If the problem asks for an exact value, keep π. If it asks for a numerical approximation, use a decimal. The calculator provides both.

The output shows a very long decimal. That is normal. Round to the desired number of decimal places as needed for your application.

Frequently Asked Questions

Why are radians used instead of degrees?

Radians simplify many mathematical formulas. For example, the arc length formula s = rθ is only valid when θ is in radians. In calculus, derivatives and integrals of trigonometric functions take their simplest form in radians.

How many radians are in a full circle?

There are 2π radians in a full circle, which is approximately 6.283185307 radians. This comes from the formula: 360° × (π/180) = 2π.

What is 1 radian in degrees?

One radian is approximately 57.2958 degrees. To convert radians to degrees, multiply by 180/π.

Can I convert negative angles?

Yes. The conversion formula works for negative degrees as well. For example, -90° converts to -π/2 radians. Just multiply the negative number by π/180.

How do I convert degrees, minutes, and seconds to radians?

First, convert the angle to decimal degrees: degrees + minutes/60 + seconds/3600. Then multiply by π/180. For example, 30° 30′ 0″ = 30.5°, which is 30.5 × π/180 ≈ 0.5323 radians.

Is it okay to round π to 3.14?

For rough estimates, yes. For precise work, use more digits. The difference can be significant in calculations involving large angles or when errors accumulate.

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in deg | Angle (degrees) | ° | 180 | 0
out rad | Radians (exact) | rad | deg * pi / 180 | 4
out rad_dec | Radians (decimal) | rad | deg * pi / 180 | 6