Degrees to Radians Conversion Formula

This calculator converts an angle given in degrees to radians. Enter the angle in degrees, and you get the equivalent value in radians, expressed as a decimal. The formula is simple: multiply the degree value by π/180.

Formula

The conversion from degrees to radians uses the relationship 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 (pi) ≈ 3.14159265358979.

Because π is an irrational number, the result is usually left in terms of π for exact mathematics, or given as a decimal for practical calculations.

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. The 180 cancels out: radians = π.
4. Thus, 180° equals π radians, which is approximately 3.1416 rad.

The calculator rounds to 4 decimal places, giving 3.1416 rad, which matches the expected result.

Why Use Radians?

Radians are the standard unit of angular measure 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 of a circle is simply s = rθ, where θ is in radians—no extra conversion factor needed. Most programming languages (like Python, JavaScript, and C) expect trigonometric functions to take radians, so converting degrees to radians is a common step in coding.

How to Choose Between Degrees and Radians

Your choice depends on the context:

• Everyday geometry and navigation – degrees are more intuitive and widely used (e.g., compass bearings, angles in construction).
• Calculus, physics, and engineering – radians are almost always required because they make derivatives and integrals cleaner.
• Computer graphics and game development – many libraries use radians internally, but you may input degrees and convert.
• Surveying and astronomy – degrees, minutes, and seconds are traditional, but radians appear in formulas.

When in doubt, check the unit expected by the formula or software you are using.

Common Mistakes and Misconceptions

• Forgetting to multiply by π – some people mistakenly use degrees × (1/180) or degrees / 180, which gives the wrong value. Always include π.
• Confusing radians with degrees in a calculator – most scientific calculators have a mode switch (DEG vs RAD). If you get a strange result, check the mode.
• Rounding too early – if you round π to 3.14 before multiplying, you lose precision. Keep more digits until the final step.
• Mixing up the conversion factor – the inverse conversion (radians to degrees) multiplies by 180/π, not π/180.

Another misconception is that radians are a “different kind� of angle. They are simply a different unit, just like inches vs. centimeters. A 90° angle is the same geometric angle as π/2 radians.

Practical Applications

Converting degrees to radians appears in many real-world tasks:

• Programming – writing code for animations, robotics, or simulations that use trigonometric functions.
• Physics – calculating angular velocity, torque, or wave functions.
• Engineering – designing gears, linkages, or structural load calculations.
• Navigation – converting between bearing angles and mathematical angles.

For example, if you are rotating a robot arm 45° and your control software expects radians, you would compute 45 × (π/180) ≈ 0.7854 rad.

Limits of the Formula

The formula works for any real number, including negative angles and angles greater than 360°. Negative degrees indicate clockwise rotation; values above 360° represent multiple full rotations. The conversion is exact because it is a linear scaling, so there is no approximation error beyond the decimal representation of π.

One limitation: the result is a decimal approximation unless you leave π as a symbol. In pure mathematics, you would often write “π/2� rather than “1.5708�. This calculator gives the decimal form for practical use.

Frequently Asked Questions

How do I convert degrees to radians without a calculator?

Multiply the degree value by π/180. For example, 30° × (π/180) = π/6. You can leave the answer in terms of π for exactness, or use 3.14159 for a decimal approximation.

What is 1 degree in radians?

1 degree equals π/180 radians, which is approximately 0.0174533 rad. This is a very small angle, about 0.0175 rad.

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 π/180.

Can I convert radians back to degrees?

Yes, multiply the radian value by 180/π. For instance, 1 radian ≈ 57.2958°. Many calculators have a built-in function for this, but the formula is simple.

Do I need to convert degrees to radians for Excel?

Excel’s trigonometric functions (SIN, COS, TAN) expect radians. If your angle is in degrees, use the RADIANS function or multiply by PI()/180. For example, =SIN(RADIANS(30)) returns 0.5.

What is the difference between degrees and radians?

Degrees divide a circle into 360 parts; radians divide it into 2π parts (about 6.283). One radian is the angle subtended by an arc equal in length to the radius. They are just different units for the same geometric angle.

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in a | Angle in degrees | ° | 180 | 0
out r | Angle in radians | rad | a * pi / 180 | 4