Angle to Radians Formula and Conversion Examples

To convert an angle from degrees to radians, multiply the degree measure by π/180: radians = degrees × π/180. This calculator takes an angle in degrees and returns its equivalent in radians, with a decimal result rounded to six places.

Formula

The conversion formula is r = d × π/180. Here, d is the angle in degrees (°), r is the same angle in radians (rad), and π is the mathematical constant approximately equal to 3.141592653589793. The second calculator result expresses the angle as a multiple of π: p = d/180, so the angle is pπ radians. For example, p = 1/2 means π/2 radians.

A radian is a unit of angle based on a circle’s radius. One radian is the angle at a circle’s center that subtends an arc whose length equals the radius. A complete circle is 2π radians, which is also 360 degrees; therefore, π radians equal 180 degrees.

Worked Example

Using the default input, d = 180°:

1. Apply the formula: r = d × π/180.
2. Substitute 180: r = 180 × π/180 = π radians.
3. As a decimal, π = 3.141592653589793…, which rounds to 3.141593 rad at six decimal places.
4. The π-multiple result is p = 180/180 = 1.000000, meaning 1π rad.

How the Conversion Works

Degrees divide a full turn into 360 equal parts. Radians measure an angle in relation to the circle’s radius, so a full turn contains 2π radians. Equating the two measures gives 360° = 2π rad. Dividing both sides by two gives the useful reference 180° = π rad, from which the conversion factor follows.

Multiplying by π/180 changes the unit without changing the angle itself. The factors cancel the degree unit, leaving radians. To go the other way, multiply radians by 180/π:

• Degrees to radians: radians = degrees × π/180.
• Radians to degrees: degrees = radians × 180/π.

For instance, 45° × π/180 = π/4 rad, or about 0.785398 rad. The exact form π/4 is often more useful than its decimal approximation in algebra and trigonometry.

Common Angle Conversions

These reference values make it easier to check a calculation or recognize common angles:

• 0° = 0 rad.
• 30° = π/6 rad ≈ 0.523599 rad.
• 45° = π/4 rad ≈ 0.785398 rad.
• 60° = π/3 rad ≈ 1.047198 rad.
• 90° = π/2 rad ≈ 1.570796 rad.
• 180° = π rad ≈ 3.141593 rad.
• 360° = 2π rad ≈ 6.283185 rad.

Negative angles convert in exactly the same way. For example, −90° equals −π/2 radians. The sign indicates direction under the usual convention; it does not change the conversion factor.

When to Use Radians

Radians are common in higher mathematics and many scientific and engineering calculations. They are especially convenient in calculus: standard derivative formulas for sine and cosine assume the angle is measured in radians. Many programming languages and mathematical libraries also expect radians for trigonometric functions, although the convention depends on the specific tool or function. Check its documentation or settings if a result seems unexpectedly wrong.

Degrees remain familiar for everyday descriptions of direction and rotation, and they are often used in geometry and introductory problems. Neither unit is inherently more accurate. Choose the one the problem, formula, instrument, or software expects, and convert before calculating if necessary.

Common Mistakes and Limits

• Using the factor backward: Degrees to radians uses π/180; radians to degrees uses 180/π.
• Leaving out π: Multiplying degrees by 1/180 alone does not produce radians. The factor must be π/180.
• Confusing an exact value with a decimal: π/3 is exact, while 1.047198 is a rounded approximation. Keep the exact form when later steps allow it.
• Mixing angle modes: A calculator or software tool set to degrees may interpret an input differently from one set to radians. Confirm the mode before evaluating trigonometric expressions.

The formula converts an angle’s unit; it does not determine an angle from other measurements or account for a particular application’s conventions. A decimal display is rounded, so results may differ slightly in the last shown digit if another tool uses a different number of decimal places or internal precision. For very large or very small angles, the same relationship still applies, but it may be helpful to preserve the input’s precision and use an exact π multiple where possible.

Frequently Asked Questions

What is the formula for converting degrees to radians?
Multiply the degree value by π/180. In symbols, r = d × π/180.

How do you convert radians to degrees?
Multiply the radian value by 180/π. For example, π/2 × 180/π = 90°.

How many radians are in 180 degrees?
180° equals π radians, approximately 3.141593 radians.

What is 90 degrees in radians?
90° equals π/2 radians, approximately 1.570796 radians.

Why do calculus formulas use radians?
Radians make many relationships between angle and arc length direct, and standard calculus identities such as the derivative of sin(x) = cos(x) use radian measure.

Do I multiply or divide by π when converting degrees to radians?
Multiply by π/180. Equivalently, divide the degree value by 180 and then multiply by π.

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in d | Angle in degrees | ° | 180 | 0
out r | Angle in radians | rad | d * pi / 180 | 6
out p | Angle in pi radians | π rad | d / 180 | 6