How to Convert Degrees to Radians

To convert degrees to radians, multiply the angle in degrees by π/180. For example, 180° equals π radians, and 90° equals π/2 radians. This calculator performs that conversion instantly: enter any angle in degrees, and it returns the equivalent in radians, using the exact value of π.

Formula

The conversion formula is:

radians = degrees × (π / 180)

Where:

• degrees – the angle you start with, measured in degrees (°)
• radians – the resulting angle, measured in radians (rad)
• π (pi) – approximately 3.14159, the ratio of a circle’s circumference to its diameter

This works because a full circle is 360° and also 2π radians. Dividing both by 2 gives 180° = π radians, which is the basis of the conversion factor π/180.

Worked Example

Using the default value of 180°:

1. Start with the formula: radians = degrees × (π / 180)
2. Substitute 180 for degrees: radians = 180 × (π / 180)
3. Cancel the 180s: radians = π
4. Numerically, π ≈ 3.1416 (rounded to 4 decimal places)

So 180° = π radians ≈ 3.1416 rad. The calculator shows this result exactly.

Why Use Radians?

Radians are the natural unit of angle in mathematics and physics. Unlike degrees, which are arbitrary (360 was chosen by ancient civilizations), radians are defined directly from the circle: one radian is the angle subtended at the center by an arc equal in length to the radius. This makes many formulas simpler. For example, the arc length formula is s = rθ only when θ is in radians. Similarly, derivatives of trigonometric functions (like d/dx sin x = cos x) assume angles are in radians. If you work in calculus, physics, or engineering, radians are almost always required.

How to Choose: Degrees vs. Radians

Which unit you use depends on context:

• Everyday geometry and navigation: degrees are common (e.g., a right angle is 90°).
• Calculus, trigonometry, and physics: radians are standard. If you see a formula like ω = 2πf, the ω is in radians per second.
• Programming and calculators: most math functions in programming languages (like sin()) expect radians. If you have degrees, convert first.
• Surveying and construction: degrees (and minutes/seconds) are typical.

When in doubt, check the unit of the answer you need. If you’re solving a problem that involves π, radians are likely intended.

Common Mistakes and How to Avoid Them

• Forgetting the π: Some people multiply by 180 only, which gives degrees, not radians. Always include π.
• Using the wrong conversion factor: To convert radians to degrees, you multiply by 180/π, not π/180. Double-check which direction you need.
• Rounding too early: If you round π to 3.14 before multiplying, you lose accuracy. Use the π button on your calculator or keep more decimals.
• Mixing units in formulas: If you plug degrees into a formula that expects radians (like the arc length formula), your answer will be wrong. Convert first.
• Negative angles: The conversion works for negative angles too. For example, -90° becomes -π/2 radians.

Common Conversions to Memorize

These appear so often that it helps to know them by heart:

• 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: divide the degree value by 180 and multiply by π. For instance, 45° / 180 = 1/4, so 45° = π/4.

Limitations of the Formula

The conversion is exact and works for any real number, including negative angles and angles larger than 360°. However, the calculator displays a decimal approximation of π, so the result is rounded to the chosen number of decimal places. For exact symbolic answers (like π/4), you need to keep π in the expression. Also, the formula assumes you are using the standard mathematical definition of radians; in some fields (like artillery) other angular units exist, but they are not radians.

Frequently Asked Questions

How do I convert radians to degrees?

Multiply the radian value by 180/π. For example, 2 radians × (180/π) ≈ 114.59°. This is the inverse of the degrees-to-radians conversion.

What is 1 radian in degrees?

1 radian ≈ 57.2958°. This comes from 180/π. It’s about 57.3°, which is useful to remember for rough estimates.

Why is π equal to 180 degrees?

Because a full circle is 360° and also 2π radians. Dividing both by 2 gives 180° = π radians. This is a definitional relationship, not a coincidence.

Can I use this calculator for negative angles?

Yes. The conversion works for any real number. For example, -45° becomes -π/4 radians. The calculator accepts negative inputs (the minimum is set to -999999).

What if I have an angle in degrees, minutes, and seconds?

First convert to decimal degrees: degrees + minutes/60 + seconds/3600. Then multiply by π/180. For example, 30°30′ = 30 + 30/60 = 30.5°, which is 30.5 × π/180 ≈ 0.5323 rad.

Do I need to use radians in Excel?

Excel’s trigonometric functions (like SIN, COS) expect radians by default. If your angle is in degrees, use the RADIANS function to convert, or multiply by PI()/180. For example, =SIN(RADIANS(30)) gives 0.5.

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in deg | Angle in degrees | ° | 180 | -999999
out rad | Angle in radians | rad | deg * pi / 180 | 4