Degrees to Radians: Formula & Conversion

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

Formula

The conversion formula is:

radians = degrees × (π / 180)

Where:

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

This formula comes from the fact that a full circle is 360° and also 2π radians. Dividing both sides by 360 gives 1° = π/180 radians, so multiplying any number of degrees by π/180 converts it to radians.

Worked Example

Using the default input of 180 degrees:

1. Start with the formula: radians = degrees × (π / 180)
2. Plug in 180 for degrees: radians = 180 × (π / 180)
3. The 180 cancels out: radians = π
4. Decimal value: π ≈ 3.1416

So 180° equals π radians (about 3.1416 rad). The calculator shows this as 3.1416 rad and the fractional form as 1π.

Why Use Radians?

Radians are the natural unit of angle 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 is only valid with radians. In engineering, computer graphics, and most calculus-based work, radians are the default. Degrees are more convenient for everyday measurements (like a 90° angle) but become awkward in higher math.

How to Convert Manually

If you need to do the conversion without a calculator:

• Multiply the degree value by π/180.
• Simplify the fraction if you want an exact answer in terms of π.
• For a decimal, replace π with 3.14159 and divide.

Example: Convert 45° to radians.

• 45 × (π/180) = 45π/180
• Simplify: divide numerator and denominator by 45 → π/4
• Decimal: 0.7854 rad

Common conversions: 30° = π/6, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

Common Mistakes and Misconceptions

• Forgetting the π: Some people write just the fraction (e.g., 0.5) instead of 0.5π. Always include π in the exact form.
• Using degrees in trig functions: When using a calculator or programming language, check whether the function expects degrees or radians. Most math libraries default to radians.
• Mixing up conversion direction: To go from radians to degrees, multiply by 180/π, not π/180.
• Rounding too early: If you round π to 3.14, you get a slightly different result. Use enough digits (3.14159) for accuracy.

Limitations of the Formula

The formula is exact and works for any real number, including negative angles and angles greater than 360°. However, the decimal output is an approximation because π is irrational. The fractional form (like 5π/6) is exact. Also, note that the calculator accepts only numeric input; if you need to convert an angle like 30°15′ (degrees and minutes), you must first convert to decimal degrees (30.25°) before using the formula.

Practical Applications

Converting degrees to radians is essential in:

• Trigonometry: Solving equations and graphing functions.
• Calculus: Derivatives and integrals of trigonometric functions.
• Physics: Angular velocity, rotational motion, wave functions.
• Engineering: Signal processing, control systems, robotics.
• Computer graphics: Rotation matrices and 3D transformations.

In all these fields, using radians avoids constant conversion factors and keeps formulas clean.

Frequently Asked Questions

How do I convert degrees to radians in Excel?

Excel has a built-in function: =RADIANS(degrees). For example, =RADIANS(180) returns 3.14159. To convert back, use =DEGREES(radians).

What is 1 radian in degrees?

1 radian is approximately 57.2958 degrees. This is because 180° = π radians, so 1 radian = 180/π degrees.

Why do we multiply by π/180?

Because a full circle is 360° and also 2π radians. Dividing both by 360 gives 1° = π/180 radians. Multiplying any degree measure by π/180 scales it to the equivalent radian measure.

Is π radians equal to 180 degrees?

Yes, exactly. This is the fundamental relationship between the two units. It means 2π radians = 360°, and π/2 radians = 90°.

Can I convert negative degrees to radians?

Yes, the formula works for negative values. For example, -90° × (π/180) = -π/2 radians. The sign indicates direction (clockwise vs counterclockwise).

What’s the difference between radians and steradians?

Radians measure plane angles in 2D. Steradians measure solid angles in 3D. A steradian is the solid angle that cuts out an area on a sphere equal to the square of the sphere’s radius. They are not interchangeable.

Related

• Tablespoons to Ounces Wet: Quick Guide
• CET Time Converter: How to Use It
• 1 Light Year in km: Exact Distance & How It’s Calculated

in deg | Angle in degrees | ° | 180 | 0
out rad | Radians (decimal) | rad | deg * (pi / 180) | 4
out frac | Radians (fraction of π) | π | deg / 180 | 3