Degrees to Radians Calculator and Conversion Formula

A degrees-to-radians calculator converts an angle you enter in degrees into its equivalent measure in radians. Enter a degree value, including a negative angle if needed, and the calculator returns radians; the conversion uses the exact factor π/180.

Formula

Multiply the angle in degrees by π/180 to convert it to radians:

r = d × π/180

• d is the input angle, measured in degrees (°).
• r is the converted angle, measured in radians (rad).
• π is the mathematical constant pi, approximately 3.141592653589793.

Degrees and radians are two ways to measure the same thing: the amount of rotation between two directions. The conversion changes the unit, not the angle itself. The inverse conversion is d = r × 180/π.

Worked Example

Using the calculator’s default input of 180°:

1. Start with r = d × π/180.
2. Substitute d = 180: r = 180 × π/180.
3. Cancel 180/180: r = π.
4. As a decimal rounded to six places, r = 3.141593 rad.

So 180° equals π radians, or approximately 3.141593 radians. The exact value is π; the decimal is rounded for display.

Why Degrees and Radians Both Exist

A full turn is 360 degrees, a familiar division used for angles in everyday diagrams and navigation. Radians are defined from a circle: one radian is the angle at the center of a circle that subtends an arc equal in length to the radius. A circle’s circumference is 2π times its radius, so a full turn measures 2π radians.

This geometric definition makes radians especially useful in mathematics and physics. In calculus, many important identities and derivatives take their simplest form when angles are in radians. For example, the derivative of sin(x) is cos(x) when x is in radians; using degrees introduces an extra conversion factor.

Common equivalents include 90° = π/2 rad, 180° = π rad, and 360° = 2π rad. These are exact relationships, not approximations.

When to Use the Conversion

Convert degrees to radians when a formula, graphing tool, programming function, or scientific calculation expects radians. Many programming languages and mathematical software use radians for trigonometric functions, but conventions can differ between tools. Check the relevant function’s documentation or angle-mode setting before entering values.

• Calculus and algebra: Use radians for trigonometric derivatives, limits, and many standard formulas.
• Geometry: Convert when a circle or rotation problem gives an angle in degrees but uses arc length or sector area formulas with radians.
• Programming: Confirm whether functions such as sine, cosine, and tangent expect radians or degrees.
• Science and engineering: Follow the units specified in the equation or measurement system being used.

Radians are technically dimensionless because they are a ratio of arc length to radius, both lengths. The label “rad” is still useful: it makes clear that an angle is being expressed in radians rather than degrees.

Reading Results and Choosing Precision

The calculator shows a decimal rounded to six places, which is convenient for many routine calculations. For exact work, keep the result in terms of π where possible. For instance, 60° is exactly π/3 rad, while its decimal form is about 1.047198 rad. Rounding too early can introduce small errors, especially when several calculations are chained together.

Use enough digits for the precision required by the problem, but do not mistake extra decimal places for greater accuracy in the original measurement. If an angle was measured only to the nearest degree, reporting a converted result to many decimal places does not make the measurement itself more precise.

Common Mistakes and Limitations

• Using the inverse factor: Degrees to radians uses π/180. Radians to degrees uses 180/π.
• Entering radians as degrees: If an angle is already in radians, do not convert it again as though it were degrees.
• Ignoring calculator mode: A handheld calculator may be set to degree or radian mode. This setting affects trigonometric evaluations, though it does not change the conversion formula itself.
• Rounding π too soon: Use the calculator’s π constant or retain π symbolically until the final step.
• Assuming angles must be positive: Negative angles are valid and represent rotation in the opposite direction under the usual sign convention. Angles greater than 360° are also valid; they can represent multiple turns.

This conversion only changes units. It does not determine an angle from a drawing, resolve which direction an angle should be measured, or reduce an angle to a particular range. For example, 450° converts to 5π/2 rad; reducing it to an equivalent angle within one full turn is a separate step.

Frequently Asked Questions

How do you convert degrees to radians?
Multiply the degree measure by π/180. For example, 45° × π/180 = π/4 rad, approximately 0.785398 rad.

How many radians are in 180 degrees?
There are exactly π radians in 180°. In decimal form, π is approximately 3.141593.

What is the formula for converting radians back to degrees?
Multiply radians by 180/π. For example, π/2 rad × 180/π = 90°.

Can I convert a negative degree value?
Yes. Apply the same formula, preserving the negative sign. For example, −90° equals −π/2 rad.

Why does my calculator give a different sine or cosine result?
Its angle mode may not match the unit you entered. Trigonometric functions interpret inputs as degrees or radians according to the calculator’s setting; conversion itself always follows the formula above.

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in d | Degrees | ° | 180 | 
out r | Radians | rad | d * pi / 180 | 6