Degrees to Radians Conversion: Formula & Examples

This calculator converts an angle from degrees to radians, or from radians to degrees, using the standard conversion factor π radians = 180°. You enter the angle value and its unit, and the calculator returns the equivalent angle in the other unit, rounded to two decimal places.

Formula

The conversion between degrees and radians is based on the relationship that a full circle is 360° and also 2π radians. Therefore:

To convert degrees to radians:
radians = degrees × (π / 180)

To convert radians to degrees:
degrees = radians × (180 / π)

Where:

• degrees – the angle in degrees (°)
• radians – the angle in radians (rad)
• π – the mathematical constant, approximately 3.141592653589793

Note that radians are dimensionless; they are a ratio of arc length to radius. In the calculator above, the unit selector multiplies the input by the appropriate factor: for degrees, it multiplies by π/180 (≈0.0174533), and for radians, it multiplies by 180/π (≈57.29578).

Worked Example

Using the default values from the calculator: angle = 45, unit = Degrees.

1. Start with the formula: radians = degrees × (π / 180)
2. Plug in the value: radians = 45 × (π / 180)
3. Simplify: radians = 45 × 0.017453292519943295
4. Calculate: radians = 0.7853981633974483
5. Round to two decimal places: 0.79 radians

The calculator shows 0.79 as the converted angle, matching this result.

Why Use Radians?

Radians are the natural unit of angle in mathematics and physics because they simplify many formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. In degrees, the derivative would include a factor of π/180. Similarly, the arc length formula s = rθ assumes θ is in radians. When you study calculus, trigonometry, or any advanced science, you will almost always work in radians.

Degrees, on the other hand, are more convenient for everyday navigation, construction, and geometry because 360 divides evenly by many numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, etc.). That makes mental arithmetic easier for common angles.

How to Choose the Right Unit

Here are some guidelines to help you decide when to use degrees vs. radians:

• Use degrees for: basic geometry, surveying, navigation, carpentry, and any application where you communicate angles to people who aren’t mathematicians.
• Use radians for: calculus, physics equations (like angular velocity ω = θ/t), signal processing, and computer graphics (most math libraries use radians).
• Check your calculator settings – many scientific calculators have a DEG/RAD mode. If you get a strange answer, check whether the mode matches your input.

If you’re unsure, convert to the unit that makes the formula you’re using simplest. For example, if you’re calculating the period of a pendulum, the formula T = 2π√(L/g) uses radians implicitly, so you should input the angle in radians if it appears in the formula.

Common Mistakes and Misconceptions

One frequent error is forgetting that π is a number, not a variable. π ≈ 3.14159, so 1 radian is about 57.3°, not 180°. Another mistake is mixing up the conversion factors: to go from degrees to radians you multiply by π/180, not 180/π. A quick check: since 180° = π radians, a smaller number of radians than degrees for the same angle (except for 0°).

Also, be careful with the word “pi� in expressions – in the calculator, π is represented by the constant pi. When you see “π/2 radians,� that is approximately 1.5708 radians, which equals 90°.

Another misconception is that radians are an arbitrary unit. In fact, they are dimensionless because they are defined as the ratio of arc length to radius. That is why they disappear in many physical equations.

Conversion Table for Common Angles

Here are some frequently used angles in both units:

• 0° = 0 rad
• 30° = π/6 ≈ 0.5236 rad
• 45° = π/4 ≈ 0.7854 rad
• 60° = π/3 ≈ 1.0472 rad
• 90° = π/2 ≈ 1.5708 rad
• 180° = π ≈ 3.1416 rad
• 270° = 3π/2 ≈ 4.7124 rad
• 360° = 2π ≈ 6.2832 rad

This table can help you quickly estimate conversions without a calculator.

Limitations of the Conversion

This conversion is exact and simple; there are no hidden limitations. However, the calculator rounds to two decimal places, so for very precise work you may want to keep more digits. Also, the input angle can be any real number, including negative angles and angles greater than 360° (or 2π radians). For example, 720° is equivalent to 2 full rotations, which is 4π radians. The conversion works the same for those values.

One thing to note: if you are working with angles in degrees-minutes-seconds (DMS) format, you must convert to decimal degrees first. For instance, 30°30′ is 30.5°.

Frequently Asked Questions

Why is 180 degrees equal to π radians?

Because a full circle is 360° and also 2π radians. Dividing both by 2 gives 180° = π radians. This is a definition, not a coincidence – radians are defined so that the arc length of a semicircle (half the circumference) equals π times the radius, and that angle is 180°.

How do I convert degrees to radians without a calculator?

Multiply the number of degrees by π/180. For example, 60° × π/180 = π/3. You can leave the answer in terms of π, or approximate with 3.14159 if you need a decimal. For common angles, memorize the fractions: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2.

What is the difference between a degree and a radian?

A degree is 1/360 of a full rotation, so it is an arbitrary division. A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. That makes radians a natural, dimensionless measure. One radian is about 57.2958°.

Can I convert radians to degrees using the same calculator?

Yes. The calculator has a unit selector. If you choose “Radians� as the unit, it will multiply the input by 180/π to give degrees. For example, entering 1 radian with unit set to Radians gives 57.30°.

Why do some formulas require radians and not degrees?

Because many mathematical identities, like the Taylor series for sin(x) or the derivative of sin(x), are only true when x is in radians. In degrees, those formulas would have extra conversion factors. Physics equations involving angular velocity and angular acceleration also assume radians to keep units consistent.

What is the symbol for radians?

Radians are often written with “rad� after the number, like 2.5 rad. Sometimes they are written without any symbol because they are dimensionless. In contrast, degrees are always written with the degree symbol (°).

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in angle | Angle value | - | 45 | 0
sel unit | Unit | Degrees=1, Radians=0.017453292519943295
out result | Converted angle | - | angle * (unit == 1 ? 0.017453292519943295 : 57.29577951308232) | 2