Angle to Radians Equation: Formula & Examples

This calculator converts an angle given in degrees to radians, and it can also convert from radians to degrees. You enter the angle value and select the unit (degrees or radians), and the calculator returns the equivalent angle in the other unit. The core equation is simple: radians = degrees × (π / 180).

Formula

The relationship between degrees and radians is defined by the fact that a full circle is 360° and also 2π radians. Therefore:

• Degrees to radians: rad = deg × (π / 180)
• Radians to degrees: deg = rad × (180 / π)

In these formulas, deg is the angle in degrees, rad is the angle in radians, and π (pi) is approximately 3.14159265. The unit of radians is dimensionless (it’s a ratio of arc length to radius), but we write “rad” to clarify. For example, 180° × (π/180) = π rad, and 2π rad × (180/π) = 360°.

Worked Example

Using the default values from the calculator above: angle = 180, unit = degrees. We want to convert 180° to radians.

1. Start with the degrees-to-radians formula: rad = deg × (π / 180).
2. Plug in the value: rad = 180 × (π / 180).
3. Simplify: the 180 in the numerator and denominator cancel, leaving rad = π.
4. Numerically, π ≈ 3.1416 (rounded to 4 decimal places).

The calculator shows 3.1416 rad, which matches. If you switch the input unit to radians and enter 3.1416, you get back 180°.

Why Use Radians?

Radians are the natural unit of angle in mathematics and physics because they simplify many formulas. For instance, the derivative of sin(x) is cos(x) only when x is in radians. In degrees, that derivative would have an extra factor of π/180. Similarly, the arc length formula s = rθ works directly only when θ is in radians. In engineering, computer graphics, and most scientific calculations, radians are the default unit. Even in everyday math, trigonometric functions in programming languages (like Python’s math.sin, JavaScript’s Math.sin) expect radians unless you convert explicitly.

Degrees are more convenient for navigation, construction, and everyday angles because 360 divides evenly into many parts (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180). But for calculus, physics, and any formula involving derivatives or integrals of trig functions, radians are essential.

Common Conversions to Memorize

• 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

These come from the basic equation. For example, 90° × (π/180) = π/2. Knowing these helps you quickly estimate conversions without a calculator.

How to Choose the Right Unit

When solving a problem, look at the context:

• If you’re using a formula from calculus, physics, or engineering (like angular velocity ω = θ/t, or the period of a pendulum), use radians.
• If you’re measuring an angle in a geometric figure or a real-world object (like a roof pitch or a road grade), degrees are common.
• If you’re using a calculator or programming language, check the default mode. Most scientific calculators have a DEG/RAD/GRAD setting. In code, use the built-in conversion functions or multiply by π/180.

When you need to convert, this calculator does it instantly. Just enter the angle and select the unit you have.

Common Mistakes and Pitfalls

• Forgetting to convert: Using degrees in a formula that expects radians (like sin in Excel) gives wildly wrong results. Always check the unit.
• Confusing π and 180: The conversion factor is π/180, not 180/π. Remember: degrees are larger, so to convert to radians you multiply by a small number (π/180 ≈ 0.01745). To convert back, multiply by a large number (180/π ≈ 57.2958).
• Rounding too early: If you round π to 3.14, you get 3.14 rad for 180°, which is off by 0.0016. For precise work, keep π in your calculator or use enough digits.
• Forgetting that radians are dimensionless: Sometimes you’ll see rad written as a unit, but it’s actually a pure number. In dimensional analysis, radians are treated as dimensionless.

Limits of the Equation

The conversion equation is exact for all real numbers, including negative angles and angles greater than 360°. For example, 720° = 4π rad. There is no rounding error if you keep π symbolic. However, when you use a decimal approximation for π, you introduce a tiny error. Also, the equation assumes you’re using the standard definition of a radian (arc length divided by radius). In some specialized fields like surveying, they use gradians (400 per circle), but that’s a different unit—not to be confused with radians.

For very large angles (like millions of degrees), the conversion still works, but you might want to reduce the angle modulo 360° first to keep it manageable. The calculator above does not automatically reduce, so it will show the full converted value.

Frequently Asked Questions

How do I convert degrees to radians in Excel?

Use the RADIANS function: =RADIANS(180) returns π (approximately 3.14159). To convert radians to degrees, use DEGREES(pi()). If you’re writing a formula manually, multiply by PI()/180.

Why is 180 degrees equal to pi 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.

What is 1 radian in degrees?

1 radian is approximately 57.2958°. To get this, use the formula: degrees = radians × (180/π). So 1 × (180/π) ≈ 57.2958°.

Can I convert angles in degrees, minutes, and seconds to radians?

Yes. First convert to decimal degrees: degrees + minutes/60 + seconds/3600. Then multiply by π/180. For example, 30° 30′ 0″ = 30.5°, and 30.5 × π/180 ≈ 0.5323 rad.

What is the difference between radians and steradians?

Radians measure plane angles (2D). Steradians measure solid angles (3D), like the field of view of a cone. One steradian is the solid angle subtended at the center of a sphere by an area on its surface equal to the square of the radius. They are not directly convertible.

Do I need to convert to radians for trigonometric functions?

Yes, in most mathematical contexts, especially calculus. The derivative and integral formulas assume radians. In practical applications like surveying, degrees are fine, but if you’re using a calculator, make sure it’s in the correct mode.

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in angle | Angle value | deg | 180 | 0
sel unit | Input unit | Degrees=1, Radians=2
out result | Converted angle | rad or deg | if(unit=1, angle*pi/180, angle*180/pi) | 4