Radians to Degrees (°) Calculator — Convert Instantly

Radians and degrees are two different ways to measure angles. In school you often see angles in degrees (like 30°, 45°, 90°), but in higher-level math, physics, engineering, computer graphics, and calculus, angles are frequently expressed in radians (like π/6, π/4, π/2). This calculator converts radians → degrees instantly — and it also shows the “nice” π-based form when possible.

The Conversion Formula

The relationship between radians and degrees comes from the geometry of a circle. A full circle is 360 degrees. That same full circle is 2π radians. Those two facts are equivalent ways of describing one complete rotation. Because they describe the same angle, we can build a ratio:

• 360° ↔ 2π radians

Divide both sides by 2π to get the conversion from radians to degrees:

degrees = radians × (180 / π)

That’s it. Multiply the radian value by 180 and divide by π. Your answer is in degrees.

Why π Is In The Formula

π is the constant that connects the circumference of a circle to its diameter. Since radians are defined using arc length on a circle, π appears naturally when you turn “how far around the circle” into a measurement of angle. A radian is literally “arc length equals radius.” When you wrap that definition around a full circle, the total arc length is the circumference (2πr), and dividing by r leaves 2π radians for one full turn.

Quick Mental Conversion Tips

If you do this often, a few common radian angles are worth memorizing. Think of π like a “unit” that maps to 180°:

• π radians = 180°
• π/2 radians = 90°
• π/3 radians = 60°
• π/4 radians = 45°
• π/6 radians = 30°
• 2π radians = 360°

From there, you can estimate others quickly. For example, 0.5 rad is a bit less than π/6 (0.5236…), so it’s a bit less than 30°.

Worked Examples

Example 1: Convert 1 rad to degrees

Use the formula: degrees = 1 × (180 / π) ≈ 57.2958°. That’s why you’ll sometimes hear “1 radian is about 57.3 degrees.”

Example 2: Convert π/2 rad to degrees

degrees = (π/2) × (180/π) = 90°. π cancels cleanly, leaving you with a neat exact value.

Example 3: Convert 2.4 rad to degrees

degrees = 2.4 × (180/π) ≈ 137.509°. Depending on your context, you might round to 137.5° or 138°.

Example 4: Negative radians

Angles can be negative (rotating clockwise instead of counterclockwise). If you convert -π/3, you get -60°. Same magnitude, opposite direction.

How This Calculator Works (Under the Hood)

When you enter a radian value, the calculator:

1. Validates the input (so empty or non-numeric entries don’t produce nonsense).
2. Computes degrees = radians × (180 / π).
3. Rounds the result to the precision you choose (for homework you may want 2–4 decimals; for code you may want more).
4. Tries to recognize common π multiples (like π/6, π/4, π/3, π/2, π, 2π) and displays a “clean” label when it’s a close match.

That “recognize π multiples” step is what makes this page feel shareable: it gives you the textbook-friendly version in plain language, even if you typed a decimal approximation.

When To Use Degrees vs Radians

You’ll see degrees more in everyday life: measuring angles with a protractor, navigation, maps, and basic geometry. Radians are more common in calculus and in many formulas involving trig functions (sin, cos, tan), because radian measure makes derivatives and integrals behave cleanly (for example, d/dx[sin x] = cos x only works as written when x is in radians).

In programming, whether you use degrees or radians depends on the library: many math libraries (JavaScript, Python, C, etc.) expect angles in radians for trig functions. So converting correctly matters a lot — a 90° angle is not 90 radians.

FAQs

• How many degrees are in 1 radian?

  1 radian ≈ 57.2958 degrees. A good quick approximation is 57.3°.

• Why does 2π radians equal 360 degrees?

  Because a full circle is one complete rotation. In radian measure, the angle equals arc length divided by radius. The full arc length is a circle’s circumference (2πr), and dividing by r gives 2π radians.

• Should I round my degree answer?

  For homework, rounding to 2–4 decimal places is usually enough unless the problem requests “exact form.” For engineering or programming, keep more precision if small errors matter.

• Can degrees be greater than 360° (or radians greater than 2π)?

  Yes. Angles can represent multiple rotations. 720° is two full turns. 4π radians is also two full turns.

• What about angles between 0° and 360°?

  If you want the “wrapped” angle, you can reduce it modulo 360°. This calculator shows the direct conversion. In many math problems, the direct value is what you want.

• Is this calculator accurate?

  Yes for the conversion — it uses the exact formula degrees = radians × (180/π). Any difference you see is from rounding or from the decimal approximation you entered.

Extra: Common Radian Values Cheat Sheet

Here’s a quick list you can copy into notes. These show up everywhere in trigonometry:

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

If you’re working on trig identities, unit circle problems, or converting for programming, this cheat sheet plus the formula will cover almost everything.

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