Degrees to Radians Conversion Factor

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 conversion factor π/180. The result is shown both as a decimal and as a multiple of π, which is the form most often used in math and physics.

Formula

The conversion factor between degrees and radians is:

radians = degrees × (π / 180)

where:

• degrees is the angle in degrees (°)
• radians is the same angle in radians (rad)
• π (pi) ≈ 3.14159265358979

This factor comes from the definition of a radian: one full circle is 360° and also 2π radians. Dividing both by 360 gives 1° = π/180 radians.

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 180s cancel: radians = π
4. As a decimal: π ≈ 3.1416 (rounded to 4 decimal places)

The calculator shows rad = 3.1416 and rad_pi = 1π, which matches the expected result.

Why Use Radians?

Radians are the natural unit of angle in mathematics 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 (radius times angle in radians), and the area of a sector is A = ½r²θ. In degrees, those formulas require extra conversion factors.

Most scientific calculators have a mode switch (DEG/RAD/GRAD) that changes how trigonometric functions interpret input. When you use calculus, physics equations, or most programming languages (like Python’s math module), angles are expected in radians by default.

Common Conversions to Memorize

Certain angles appear so often that it is worth memorizing their radian values:

• 0° = 0 rad
• 30° = π/6 rad
• 45° = π/4 rad
• 60° = π/3 rad
• 90° = π/2 rad
• 180° = π rad
• 270° = 3π/2 rad
• 360° = 2π rad

Notice the pattern: 180° always equals π, so any degree measure can be converted by multiplying by π/180 and simplifying if possible.

How to Convert Without a Calculator

If you need to convert mentally, you can use the fact that 1° ≈ 0.01745 radians. For rough estimates, multiply the number of degrees by 0.0175. For example, 57° × 0.0175 ≈ 0.9975 rad, which is close to 1 rad (the exact value is about 0.9948).

For exact values, keep π in the answer. Instead of writing 45° as 0.7854, write π/4. This is often preferred in textbooks and engineering because it is exact and easier to simplify in equations.

Common Mistakes and Pitfalls

The most frequent error is forgetting to switch your calculator to radian mode when working with trigonometric functions. If you compute sin(90) in degree mode you get 1, but in radian mode you get about 0.894. Always check the mode before starting.

Another mistake is using the reciprocal factor. To convert radians back to degrees, you multiply by 180/π, not π/180. Mixing these up gives wildly wrong answers.

Rounding too early is also a problem. If you use 3.14 instead of π, your result will be off by about 0.05%, which might matter in precise calculations. Keep π as a symbol until the final step.

Alternatives to Degrees

Degrees are not the only alternative to radians. Gradians (or grads) divide a right angle into 100 parts, so 90° = 100 grads. They are used in some surveying and engineering contexts, but rarely elsewhere. The conversion factor for degrees to grads is (10/9) per degree.

There is also the turn, where 1 turn = 360° = 2π radians. Turns are sometimes used in computer graphics and animation to represent full rotations. The conversion is simple: 1 turn = 2π radians.

For most mathematical work, radians are the standard. Degrees are more convenient for everyday geometry and navigation because 360 divides evenly by many numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180).

Frequently Asked Questions

Why is 180° 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 fundamental relationship that defines the conversion factor.

How do I convert degrees to radians in Excel?

Excel has a built-in function: =RADIANS(angle) converts degrees to radians. Conversely, =DEGREES(angle) converts radians to degrees. For example, =RADIANS(180) returns approximately 3.14159.

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 at the center of a circle by an arc equal in length to the radius. One radian is about 57.2958 degrees.

Can I convert degrees to radians by multiplying by 0.0175?

Yes, that is a good approximation because π/180 ≈ 0.0174533. For most everyday purposes, 0.0175 is close enough, but for precise work use the exact factor π/180.

Why do some formulas require radians instead of degrees?

Because the derivatives and integrals of trigonometric functions are simple only when angles are measured in radians. For example, the limit of sin(x)/x as x approaches 0 is 1 only if x is in radians. Using degrees would introduce a constant factor of π/180.

What is the conversion factor for radians to degrees?

The inverse factor is 180/π ≈ 57.2958. So to convert radians to degrees, multiply by 180/π. For instance, 2 radians × (180/π) ≈ 114.59°.

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in deg | Angle in degrees | ° | 180 | 0
out rad | Radians (decimal) | rad | deg * pi / 180 | 4
out rad_pi | Radians as multiple of π | π | deg / 180 | 2