Trigonometry Degrees to Radians: How to Convert

This calculator converts an angle from degrees to radians instantly. You enter the angle in degrees, and it returns the equivalent value in radians, which is the standard unit of angular measure in mathematics and physics. The conversion is exact and follows the simple ratio that 180° equals π radians.

Formula

The conversion from degrees to radians uses the fundamental relationship between the two units:

radians = degrees × (π / 180)

Where:

• degrees – the angle you want to convert, measured in degrees (°).
• radians – the resulting angle, measured in radians (rad).
• π (pi) – approximately 3.14159265, the ratio of a circle’s circumference to its diameter.

Because 180° equals π radians, multiplying any degree measure by π/180 scales it to the radian system.

Worked Example

Using the default value of 45°:

1. Start with the formula: radians = degrees × (π / 180)
2. Plug in 45: radians = 45 × (π / 180)
3. Simplify the fraction: 45/180 = 1/4
4. So radians = π/4 ≈ 0.7854

The calculator shows 0.7854 rad, which matches π/4 exactly. This is a common angle in trigonometry, often written as π/4.

Why Radians Matter

Radians are the natural unit for angles in higher mathematics. Unlike degrees, which are an arbitrary division of a circle into 360 parts, radians are defined directly from the circle itself: one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. This makes many formulas simpler. For example, the derivative of sin(x) is cos(x) only when x is in radians. In degrees, the derivative would involve an extra factor of π/180. Similarly, the arc length formula s = rθ and the area of a sector A = ½r²θ are clean only in radians. Engineers, physicists, and computer scientists use radians almost exclusively in calculations, which is why converting from degrees is so common.

How to Convert Manually

If you need to convert without a calculator, follow these steps:

1. Write down the angle in degrees.
2. Multiply it by π/180. You can do this in two parts: divide the degrees by 180, then multiply by π.
3. Simplify the fraction if possible (e.g., 90° → 90/180 = 1/2 → π/2).
4. If you need a decimal, multiply by 3.14159.

For common angles, memorize these exact values:

• 0° → 0
• 30° → π/6
• 45° → π/4
• 60° → π/3
• 90° → π/2
• 180° → π
• 270° → 3π/2
• 360° → 2π

Common Mistakes and Misconceptions

A frequent error is forgetting that π is a number, not a variable. When you see “π/2”, it means approximately 1.5708, not an expression with a symbol. Another mistake is mixing up the conversion direction: to go from radians to degrees, you multiply by 180/π, not π/180. Also, many students wonder why degrees are even used if radians are more “natural.” Degrees are practical for everyday measurements, navigation, and construction because 360 divides evenly by many integers (2, 3, 4, 5, 6, 8, 9, 10, 12, etc.), making mental arithmetic easier. But for calculus, physics, and any formula involving periodic functions, radians are essential.

When to Use Degrees vs. Radians

Choose the unit based on context:

• Use degrees for navigation (compass bearings), construction (angles of roofs, stairs), and most everyday geometry problems.
• Use radians for calculus, trigonometry identities, physics (angular velocity, wave functions), and any scientific computation.
• Check your calculator mode – most scientific calculators have a DEG/RAD setting. If you enter sin(90) in radian mode, you get 0.8939, not 1.

In programming, math libraries (like Python’s math module or JavaScript’s Math object) expect radians. If you have an angle in degrees, you must convert it before using sin, cos, or tan.

Limitations of the Conversion

Converting degrees to radians is a linear scaling, so it works for any real number, including negative angles and angles greater than 360°. However, the result is often an irrational number because π is irrational. The calculator rounds to 4 decimal places, which is sufficient for most practical purposes. For exact symbolic answers (like π/4), you would keep the π in the expression. Also, note that the conversion factor is exact – there is no rounding error in the formula itself, only in the decimal representation.

Frequently Asked Questions

How many radians are in 180 degrees?

Exactly π radians. This is the defining relationship: 180° = π rad. So 360° = 2π rad, and 90° = π/2 rad.

What is 1 degree in radians?

1° = π/180 ≈ 0.0174533 radians. This is a very small angle, about 0.0175 rad.

Why do we multiply by π/180?

Because 180° corresponds to π radians, the ratio of radians to degrees is π/180. Multiplying any degree measure by this ratio converts it to radians.

How do I convert radians to degrees?

Multiply the radian value by 180/π. For example, 2 radians ≈ 2 × 57.2958 = 114.5916°.

What is the difference between a radian and a degree?

A degree is 1/360 of a full circle, while a radian is the angle at which the arc length equals the radius. One full circle is 360° or 2π radians, so 1 radian ≈ 57.2958°.

Do I need to convert degrees to radians for trigonometry?

In pure mathematics, yes – most trigonometric identities and derivatives assume radians. In practical applications like surveying or architecture, degrees are common, but you must convert when using formulas that involve calculus or physics.

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in deg | Angle in degrees | ° | 45 | 0
out rad | Angle in radians | rad | deg * pi / 180 | 4