Convert Degrees to Radians: Formula & Calculator

This calculator converts an angle from degrees to radians instantly. You enter the angle in degrees, and it returns the equivalent value in radians, along with the step-by-step calculation. The formula is simple: multiply degrees by π/180.

Formula

The relationship between degrees and radians is based on the fact that a full circle is 360° and also 2π radians. Therefore:

radians = degrees × (π / 180)

Where:

• degrees – the angle you want to convert, in degrees (°)
• radians – the resulting angle, in radians (rad)
• π – the mathematical constant, approximately 3.14159265

To convert back: degrees = radians × (180 / π)

Worked Example

Using the default value of 180°:

1. Start with the formula: radians = degrees × (π / 180)
2. Plug in 180°: radians = 180 × (π / 180)
3. Simplify: the 180 cancels out, leaving radians = π
4. Numerically: π ≈ 3.1416 radians

The calculator shows 3.1416 rad (rounded to 4 decimals) and the reverse check shows 180.00°.

Why Use Radians?

Radians are the natural unit of angle in mathematics and physics. They simplify many formulas because they are dimensionless – a radian is defined as the ratio of arc length to radius. For example, the derivative of sin(x) is cos(x) only when x is in radians. In degrees, that derivative would involve an extra factor of π/180. Similarly, the arc length of a circle is simply s = rθ (with θ in radians), and the area of a sector is A = ½r²θ. Degrees are convenient for everyday measurements, but radians are essential for calculus, trigonometry, and most scientific work.

How to Convert Manually

If you need to convert without a calculator, follow these steps:

• Multiply the degree value by π (≈3.14159).
• Divide the result by 180.
• Simplify the fraction if possible (e.g., 90° → π/2).

For example, 45°:

• 45 × π = 45π
• 45π / 180 = π/4
• So 45° = π/4 rad ≈ 0.7854 rad

Common conversions you might memorize:

• 0° = 0 rad
• 30° = π/6 rad
• 45° = π/4 rad
• 60° = π/3 rad
• 90° = π/2 rad
• 180° = π rad
• 360° = 2π rad

Common Mistakes and Pitfalls

One frequent error is forgetting to multiply by π. Some people mistakenly think that 180° equals 1 radian – that is wrong; 180° equals π radians (about 3.1416). Another mistake is mixing up the conversion direction: to convert radians to degrees, multiply by 180/π, not π/180. Also, be careful with decimal precision – when you round π too early, the final answer can be off. For instance, using 3.14 instead of 3.14159 changes 180° from 3.1416 rad to 3.14 rad, a small but noticeable error in precise calculations. Finally, remember that radians are often expressed as multiples of π; leaving the answer as π/2 is usually preferred over 1.5708 in math contexts.

Practical Applications

Radians appear in many real-world fields:

• Physics: Angular velocity (ω) is measured in radians per second, and rotational kinetic energy uses radians.
• Engineering: Signal processing and control systems use radians for phase angles and frequencies.
• Computer Graphics: Rotation matrices and 3D transformations use radians in most programming libraries.
• Navigation: GPS and aviation use degrees, but conversion to radians is often needed for calculations.
• Mathematics: Trigonometric functions in calculus and Fourier transforms are defined in radians.

Even if your work uses degrees, you will often need to convert to radians when plugging into formulas or writing code.

Limitations of the Formula

The conversion formula is exact and has no inherent limitations – it works for any real number, including negative angles and angles greater than 360°. However, the calculator rounds to 4 decimal places, so for extremely precise work you may want to keep π symbolic. Also, remember that the formula assumes the degree measure is in the standard decimal system; if you have degrees, minutes, and seconds (DMS), you must first convert to decimal degrees (e.g., 30°15′ = 30 + 15/60 = 30.25°).

Frequently Asked Questions

How do I convert degrees to radians in Excel?

In Excel, use the RADIANS function: =RADIANS(180) returns π (≈3.14159). To convert radians to degrees, use =DEGREES(radians).

Why is 180 degrees equal to pi radians?

Because a full circle is 360° and also 2π radians. Dividing both by 2 gives 180° = π radians. This is the fundamental relationship that defines the conversion factor.

What is 1 degree in radians?

1 degree equals π/180 radians, approximately 0.0174533 radians. This is the conversion factor you multiply by.

Can I convert negative degrees?

Yes, the formula works for negative angles. For example, -90° converts to -π/2 radians (≈ -1.5708 rad).

Do I need to use radians in a calculator?

Most scientific calculators have a mode setting (DEG, RAD, GRAD). When doing calculus or physics, set it to RAD if your formula expects radians. For simple geometry with degrees, keep it in DEG mode.

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

First convert DMS to decimal degrees: degrees + minutes/60 + seconds/3600. Then multiply by π/180. For example, 30°15’0″ = 30 + 15/60 = 30.25°, then 30.25 × π/180 ≈ 0.5279 rad.

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in deg | Angle in degrees | ° | 180 | 0
out rad | Angle in radians | rad | deg * (pi / 180) | 4
out deg2 | Angle in degrees (check) | ° | rad * (180 / pi) | 2