Degrees to Radians Conversion Explained

Yes, degrees can be converted to radians. The conversion is a simple multiplication: multiply the angle in degrees by π/180 to get the angle in radians. This article explains the relationship between the two units, shows you how to do the conversion by hand and with a calculator, and covers common mistakes and practical examples.

Formula

The conversion formula is:

radians = degrees × (π / 180)

Where:

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

Because a full circle is 360° and also 2π radians, dividing 360° by 2π gives the conversion factor 180/π degrees per radian, or equivalently π/180 radians per degree.

Worked Example

Using the default value of 180 degrees:

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

The calculator shows the same result: 3.1416 rad (rounded to 4 decimal places).

Why Are There Two Units for Angles?

Degrees are the older, everyday unit, likely derived from ancient Babylonian astronomy where a year was approximated as 360 days. Each day the sun moved about one degree along the ecliptic. Degrees are convenient for describing everyday angles: a right angle is 90°, a straight line is 180°, and a full turn is 360°.

Radians are the natural unit in mathematics and physics because they simplify many formulas. For example, the arc length of a circle is simply s = rθ when θ is in radians, and the derivative of sin(x) is cos(x) only when x is in radians. In calculus, trigonometric functions are defined with radians to make these relationships clean.

How to Convert Degrees to Radians

To convert any angle from degrees to radians:

• Multiply the degree value by π/180.
• Simplify the fraction if possible (e.g., 90° × π/180 = π/2).
• If you need a decimal, divide the numerator by the denominator and multiply by π.

For example:

• 30° = 30 × π/180 = π/6 ≈ 0.5236 rad
• 45° = 45 × π/180 = π/4 ≈ 0.7854 rad
• 60° = 60 × π/180 = π/3 ≈ 1.0472 rad
• 270° = 270 × π/180 = 3π/2 ≈ 4.7124 rad

Common Mistakes and Misconceptions

A frequent error is confusing the conversion factor. Some people multiply by 180/π instead of π/180, which would give a huge number. Always check: a full circle is 360° = 2π ≈ 6.2832 rad, so a degree should be a small fraction of a radian (1° ≈ 0.01745 rad).

Another mistake is forgetting to include π in the result. When you convert 90°, the answer is π/2, not 1.5708 (which is the decimal approximation). In many math contexts, leaving π in the answer is preferred because it is exact.

Also, be careful with calculator modes. If your calculator is in degree mode and you enter sin(90), you get 1. But if you enter sin(1.5708) while still in degree mode, you get a different number because the calculator interprets 1.5708 as degrees. Always switch to radian mode when working with radians.

When to Use Degrees vs. Radians

Use degrees for:

• Navigation, surveying, and construction where angles are traditionally measured in degrees.
• Everyday language: “a 45-degree angle” is more common than “π/4 radians.”
• Geometry problems that specify degrees.

Use radians for:

• Calculus, trigonometry, and physics formulas.
• Graphing trigonometric functions (the x-axis is usually in radians).
• Angular velocity and frequency (e.g., ω in rad/s).

How to Convert Radians Back to Degrees

To go the other way, multiply the radian value by 180/π. For example:

• π radians = π × 180/π = 180°
• π/2 radians = (π/2) × 180/π = 90°
• 1 radian ≈ 57.2958°

This is the inverse of the degrees-to-radians conversion.

Frequently Asked Questions

Why is 180 degrees equal to π radians?

A full circle is 360° and also 2π radians. Therefore half a circle is 180° and also π radians. This is the basis of the conversion factor: 180° = π rad.

How do I convert degrees to radians in Excel?

Excel has a built-in function: =RADIANS(degrees). For example, =RADIANS(180) returns 3.14159265358979. If you need to convert radians back to degrees, use =DEGREES(radians).

What is 1 degree in radians?

1 degree is equal to π/180 radians, which is approximately 0.0174532925 radians. This is a very small angle because a full circle of 360° is about 6.2832 radians.

Can I convert degrees to radians using a calculator?

Yes, most scientific calculators have a mode switch between degrees and radians. You can enter the angle in degrees, switch to radian mode, and the calculator will display the value in radians. Alternatively, multiply by π/180 manually.

Why do we use radians instead of degrees in calculus?

Because radian measure makes the derivative formulas simple. For example, the derivative of sin(x) is cos(x) only when x is in radians. If x were in degrees, the derivative would include a factor of π/180, making formulas messy.

Is a radian a unit?

Yes, but it is a dimensionless unit. A radian is defined as the ratio of arc length to radius, so it is a pure number. That is why radians are often omitted in mathematical notation (e.g., sin(1) means 1 radian).

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in deg | Angle in degrees | ° | 180 | 0
out rad | Angle in radians | rad | deg * pi / 180 | 4
out rev | Angle in revolutions | rev | deg / 360 | 4