Convert Degrees to Radians

To convert degrees to radians, use this rule: multiply the degree value by π/180.

Formula (degrees → radians):
[ \text{radians}=\text{degrees}\times \frac{\pi}{180} ]

Quick steps:

1. Take your degree number.
2. Multiply by π/180.
3. Reduce the fraction if you can.

Example: Convert 60° to radians
[ 60\times \frac{\pi}{180}=\frac{60}{180}\pi=\frac{\pi}{3} ]

Common degree to radian conversions:

• 30° = (\pi/6)
• 45° = (\pi/4)
• 90° = (\pi/2)
• 180° = (\pi)
• 360° = (2\pi)

Use this method any time you need a clear degrees to radians conversion for trig, geometry, or the unit circle.

What is the degrees to radians conversion formula?

Use this formula to convert degrees to radians:

radians = degrees × (π / 180)

Since π radians equals 180 degrees, the conversion always uses π/180.

How do you convert radians to degrees?

Use the reverse formula:

degrees = radians × (180 / π)

This helps when a value is given in radians and you need degrees for angles or bearings.

Why do we convert degrees to radians?

Radians are the standard unit for many math and physics formulas. Most trig formulas work best in radians, especially in calculus. You also see radians in engineering, motion, waves, and rotations.

What are the most common degree to radian conversions?

Here are popular angle conversions:

• 0° = 0 rad
• 30° = π/6 rad
• 45° = π/4 rad
• 60° = π/3 rad
• 90° = π/2 rad
• 120° = 2π/3 rad
• 135° = 3π/4 rad
• 150° = 5π/6 rad
• 180° = π rad
• 270° = 3π/2 rad
• 360° = 2π rad

How many radians are in a full circle?

A full circle is 360°, which equals 2π radians.
A half circle is 180°, which equals π radians.

What is 1 degree in radians?

1° = π/180 radians, which is about 0.0174533 rad.

This is useful when you need a quick estimate for small angles.

What is 1 radian in degrees?

1 rad = 180/π degrees, which is about 57.2958°.

A radian is a bit less than 60 degrees.

How do you convert degrees to radians without a calculator?

Keep π as a symbol and reduce the fraction:

1. Write the angle as a fraction over 1: degrees/1
2. Multiply by π/180
3. Reduce the fraction

Example: 150° × π/180 = 150π/180 = 5π/6

How do you convert negative degrees to radians?

Convert the same way, then keep the negative sign:

radians = (negative degrees) × (π / 180)

Example: -45° = -45π/180 = -π/4

Are degrees and radians the same thing?

No. They measure angles using different scales.

• Degrees split a circle into 360 parts.
• Radians measure angles using the circle’s radius. One radian is the angle that makes an arc length equal to the radius.

Both are correct; the best choice depends on the task.

Do trig functions like sin and cos use degrees or radians?

Trig functions can use either, but many math courses and most higher math use radians. If you’re entering values into software or code, radians are often the default. Always check what unit the input expects.

Why does π show up in the degrees to radians conversion?

π connects angles to circles. A circle’s circumference is 2πr, and the radian comes from arc length divided by radius. Since 180° matches half a circle, it equals π radians, which makes π part of the conversion.

How do you convert degrees to radians in terms of π?

Multiply by π/180 and simplify, leaving π in the result.

Example:
225° × π/180 = 225π/180 = 5π/4

This format is common in geometry and trigonometry.

What is the fastest way to estimate degrees to radians?

Use this shortcut:

radians ≈ degrees × 0.01745

Example:
60° ≈ 60 × 0.01745 ≈ 1.047 rad

For exact work, keep π/180 instead of rounding.

When should you use radians instead of degrees?

Use radians when you work with:

• calculus problems (derivatives and integrals of trig functions)
• angular speed and rotation formulas
• wave motion and phase angles
• physics and engineering equations that include π

Degrees are often used for basic geometry, bearings, and everyday angle measures.