Convert Angle to Radians: Formula & Examples

This calculator converts an angle given in degrees to radians, and also shows the reverse conversion from radians to degrees. You enter the angle value and its unit (degrees or radians), and the calculator returns the equivalent angle in the other unit. The conversion uses the fixed relationship that 180° equals π radians.

Formula

The core formula for converting degrees to radians is:

radians = degrees × (π / 180)

And for converting radians to degrees:

degrees = radians × (180 / π)

Where:

• degrees – the angle in degrees (°), a dimensionless number but conventionally written with the degree symbol.
• radians – the angle in radians (rad), also dimensionless but often written with “rad” for clarity.
• π (pi) – approximately 3.14159265358979, the ratio of a circle’s circumference to its diameter.

The factor 180/π comes from the definition of a radian: one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. A full circle is 360°, which equals 2π radians, so half a circle (180°) equals π radians.

Worked Example

Using the calculator’s default values: angle = 45°, unit = Degrees.

Step 1: Write the conversion formula.
radians = degrees × (π / 180)

Step 2: Plug in 45 for degrees.
radians = 45 × (π / 180)

Step 3: Simplify the fraction 45/180 = 1/4.
radians = π / 4

Step 4: Compute the decimal value. π/4 ≈ 0.7853981633974483, which rounds to 0.7854.

The calculator shows 0.7854 rad, matching the result. The reverse output (if you entered radians) would use the same relationship.

Why Use Radians Instead of Degrees?

Radians are the natural unit for angles in mathematics and physics because they simplify many formulas. In calculus, the derivative of sin(x) is cos(x) only when x is in radians; if x were in degrees, an extra factor of π/180 would appear. Similarly, the arc length formula s = rθ and the area of a sector A = ½r²θ are valid only with radians. Degrees are convenient for everyday navigation and construction, but radians are essential for higher-level math, engineering, and scientific computing.

Common Conversions and Examples

Here are some frequently used degree-to-radian conversions:

• 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

To convert a negative angle, the same formula applies. For example, -30° = -π/6 ≈ -0.5236 rad. Angles can be any real number; they are not limited to 0–360°.

How to Convert Manually (Step-by-Step)

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

1. Identify the angle value and its unit. If it’s in degrees, proceed to step 2. If it’s in radians, use the reverse formula (step 4).
2. Multiply the degree value by π/180. You can also multiply by π first, then divide by 180.
3. Simplify the fraction if possible. For example, 60° × π/180 = 60π/180 = π/3.
4. For radians to degrees, multiply the radian value by 180/π. For example, 2 rad × 180/π ≈ 114.59°.

Remember that π is an irrational number; using 3.14159 is usually sufficient for hand calculations.

Common Mistakes and How to Avoid Them

One frequent error is mixing up the conversion factors. Always check: degrees → radians uses π/180; radians → degrees uses 180/π. Another mistake is forgetting to include π in the result – many students write 45 as “45 rad” instead of “π/4 rad”. Also, when using a calculator, ensure it is in the correct mode (degree or radian) for the trigonometric functions you are evaluating. The angle conversion itself does not depend on calculator mode, but subsequent calculations do.

Limitations and Practical Notes

The conversion is exact for any real number, but the decimal output is rounded to a limited number of digits. For most practical purposes, four decimal places are sufficient. In advanced work, keep the exact expression (e.g., π/4) rather than a decimal approximation. The calculator works with any positive or negative angle, and it does not require the angle to be within a specific range – angles can be greater than 360° or less than 0°.

Frequently Asked Questions

How many radians are in 180 degrees?

Exactly π radians. This is the fundamental relationship used in the conversion formula.

What is 1 radian in degrees?

1 radian ≈ 57.2958°. This comes from dividing 180° by π.

Why do we use 180/π for conversion?

Because a full circle is 360° and also 2π radians, so 180° = π radians. Dividing both sides by 180 gives 1° = π/180 radians; dividing by π gives 1 radian = 180/π degrees.

Can I convert degrees to radians without π?

No, π is inherently part of the relationship between degrees and radians. The conversion always involves π, but you can use a decimal approximation like 3.14159 for practical calculations.

Is a radian a unit?

Yes, but it is a dimensionless unit. Radians measure the ratio of arc length to radius, so they have no physical dimension. This is why the SI unit for angle is the radian.

How do I convert degrees to radians in Excel?

Use the RADIANS function: =RADIANS(45) returns 0.785398163. The DEGREES function does the reverse: =DEGREES(0.785398163) returns 45.

Related

• Control Resonant: Does the Evacuation Zone 1 Kit Unlock Its Map?
• K Constant in Physics: Coulomb & Spring Meaning
• Tablespoons to Ounces Wet: Quick Guide

in a | Angle value | - | 45 | 0
sel u | Unit | Degrees=1, Radians=2
out r | Radians | rad | a * pi / 180 | 4
out d | Degrees | deg | a * 180 / pi | 2