Degrees to Radians Formula & Examples

This calculator converts an angle given in degrees to radians. You enter the angle in degrees, and it returns the equivalent value in radians, using the formula radians = degrees × (π / 180). The result is shown as a decimal number, and you can also see the exact fractional form with π in the explanation below.

Formula

The conversion from degrees to radians is based on the fact that a full circle is 360 degrees and also 2π radians. Therefore:

radians = degrees × (π / 180)

Where:

• degrees – the angle you want to convert, in degrees (°).
• radians – the equivalent angle in radians (rad), a dimensionless unit.
• π (pi) ≈ 3.141592653589793…

This formula works for any real number, including negative angles and angles greater than 360°.

Worked Example

Using the default value of 180 degrees:

1. Start with the formula: radians = degrees × (π / 180).
2. Plug in 180 for degrees: radians = 180 × (π / 180).
3. Simplify: 180 / 180 = 1, so radians = 1 × π.
4. Result: radians = π ≈ 3.1416.

The calculator shows 3.1416 rad, which matches π. For a quick check, 90° gives π/2 ≈ 1.5708, and 360° gives 2π ≈ 6.2832.

Why Convert Degrees to Radians?

Radians are the natural unit for angles in mathematics and physics because they simplify many formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. Similarly, the arc length formula s = rθ requires θ in radians. Without the conversion, you would need extra factors of π/180 in every calculation.

In practical fields like engineering, computer graphics, and navigation, most internal calculations use radians, even if the user interface shows degrees. Understanding the conversion helps you interpret results and debug code.

How to Choose: Degrees vs. Radians

Use degrees when you are dealing with everyday angles, such as in construction, surveying, or simple geometry problems. Use radians when you are doing calculus, trigonometry identities, Fourier transforms, or any physics involving rotational motion. Many scientific calculators have a mode switch (DEG/RAD) – always check which mode is active before entering an angle.

If a problem gives an angle in degrees but the formula expects radians, convert first. For example, the sine of 30° is 0.5, but sin(30) in radian mode is −0.988, a common mistake.

Common Mistakes and How to Avoid Them

• Forgetting the π factor: Multiplying by 180 instead of π/180 gives a huge wrong number. Remember: 180° = π rad, so the conversion factor is π/180.
• Mixing up degrees and radians in calculator mode: Always check whether your calculator is in DEG or RAD mode. The same number 30 will give different results.
• Rounding too early: If you round π to 3.14, the result may be off in the third decimal. Use the full precision of your calculator or the π button.
• Confusing degrees with gradians: Some calculators also have a GRAD mode (400 grads per circle). Make sure you are not accidentally in that mode.

Limitations of the Formula

The formula is exact and works for all real numbers, but the output as a decimal is an approximation because π is irrational. For exact symbolic results, you would keep π in the answer (e.g., 45° = π/4 rad). The calculator gives a decimal approximation, which is fine for most practical uses. For extremely large angles, the decimal may have many digits, but the conversion remains accurate.

Frequently Asked Questions

How many radians are in 1 degree?

One degree is equal to π/180 radians, approximately 0.0174533 radians. This is a small angle, which is why a full circle (360°) is about 6.2832 radians.

Why is 180 degrees equal to π radians?

Because a full circle is 360° and also 2π radians. Dividing both by 2 gives 180° = π radians. This is the basis of the conversion factor.

How do I convert radians back to degrees?

Multiply the radians by 180/π. For example, π/2 radians × (180/π) = 90°. Many calculators have a built-in conversion function, but the formula is simple.

What is the difference between a degree and a radian?

A degree is 1/360 of a full circle, while a radian is the angle subtended by an arc equal in length to the radius. Radians are dimensionless and more natural for mathematical analysis.

Can I convert negative degrees to radians?

Yes, the formula works for negative values. For example, −90° × (π/180) = −π/2 ≈ −1.5708 radians. Negative angles just indicate direction.

Do I need to convert degrees to radians for trigonometric functions?

In calculus and most mathematical contexts, yes, because the standard derivative and series formulas assume radians. In practical geometry, you can use degrees as long as you are consistent, but check the mode on your calculator.

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in deg | Angle in degrees | deg | 180 | 0
out rad | Angle in radians | rad | deg * pi / 180 | 4