How to Convert Degrees to Radians

Type sin(30) into most programming languages and you won't get 0.5. In Python, JavaScript, C and Java, you'll get about −0.988.

That isn't a bug. These languages expect angles in radians, not degrees, and 30 radians is almost five full turns around a circle. Converting degrees to radians is one of the most common small tasks in programming, physics and trigonometry, and one of the most common sources of mistakes.

Radians Aren't Arbitrary, Degrees Are

Degrees feel natural because we learn them first. But 360 is an arbitrary choice, likely inherited from ancient Babylonian astronomy and its base-60 number system.

Radians come directly from the circle itself. One radian is the angle where the arc length equals the radius. There's nothing arbitrary about it, which is why calculus formulas only work cleanly in radians.

The Formula

Radians = Degrees × π / 180

Why π/180? A full circle is 360°. Its circumference is 2πr, so a full circle measures 2π radians:

360° = 2π radians
180° = π radians
1°   = π/180 radians ≈ 0.0174533 radians

Convert instantly with the degrees to radians calculator.

Step-by-Step Examples

Example 1: 30°

30 × π / 180 = 30π / 180 = π/6 ≈ 0.5236 radians

Example 2: 135°

135 × π / 180 = 135π / 180

Simplify the fraction by dividing top and bottom by 45 (their GCF):

= 3π/4 ≈ 2.3562 radians

Use the GCF calculator to simplify these fractions quickly.

Example 3: 270°

270 × π / 180 = 3π/2 ≈ 4.7124 radians

Example 4: A Non-Standard Angle, 47°

47 × π / 180 ≈ 47 × 0.0174533 ≈ 0.8203 radians

Conversion Table for Common Angles

A quick memory trick: 45° = π/4, and everything else in the first quadrant is a multiple or fraction of that.

Converting Degrees, Minutes and Seconds

Angles in navigation and astronomy are often written like 40° 26′ 46″. First convert to decimal degrees:

40 + 26/60 + 46/3600 ≈ 40.4461°

Then convert to radians:

40.4461 × π / 180 ≈ 0.7059 radians

The DMS converter handles the first step.

An Insider Reference: Who Named the Radian?

The idea of measuring angles by arc length relative to radius goes back to English mathematician Roger Cotes, who described it around 1714. Leonhard Euler used the approach extensively.

The word "radian" came much later. It's generally traced to James Thomson, an engineer and brother of physicist Lord Kelvin, who used it in examination questions at Queen's College, Belfast, in 1873. Mathematician Thomas Muir had been considering names like "rad" and "radial." "Radian" won, and in 1960 it became part of the International System of Units (SI).

Why Radians Matter

Arc Length Becomes Simple

In radians, the length of an arc is just:

s = r × θ

In degrees, you'd need s = r × θ × π/180.

Calculus Only Works Cleanly in Radians

The derivative of sin(x) is cos(x) only when x is in radians. In degrees, an awkward π/180 factor appears everywhere. See the calculus formulas.

Small-Angle Approximation

For small angles in radians, sin(θ) ≈ θ. For 0.1 radians, sin(0.1) ≈ 0.0998. Engineers and physicists use this constantly, for example in pendulum equations.

Common Mistakes

1. Mixing modes on a calculator. Scientific calculators have a DEG/RAD setting. If sin(90) doesn't give 1, you're probably in radian mode.

2. Converting twice. Some functions already accept degrees. Converting an angle that's already in radians shrinks it by a factor of about 57.

3. Rounding π too early. Using 3.14 instead of a more precise value adds an error of about 0.05%. For exact answers, keep π symbolic, like 3π/4, and only convert to a decimal at the end.

4. Forgetting negative and large angles. The formula works for any angle: −90° is −π/2, and 720° is 4π, two full turns.

Two Concepts Worth Knowing

Radian

A radian is the angle that cuts off an arc equal in length to the circle's radius. One radian ≈ 57.2958°.

Unit Circle

The unit circle has radius 1, so arc length equals the angle in radians. Every point on it is (cos θ, sin θ). See the trigonometric formulas.

Quick Answer: How Do You Convert Degrees to Radians?

Multiply the angle in degrees by π/180. For example, 30° × π/180 = π/6 ≈ 0.5236 radians, and 135° × π/180 = 3π/4 ≈ 2.3562 radians. A full circle is 360° or 2π radians, and 180° equals π radians.

Try Them Yourself

• Degrees to Radians Calculator: convert any angle instantly
• Radians to Degrees Calculator: go the other way
• DMS Converter: degrees, minutes and seconds
• Angle Unit Converter: degrees, radians and gradians
• Trigonometric Tables: sine and cosine for common angles
• How to Convert Radians to Degrees: the reverse conversion

Open a programming console and compare Math.sin(30) with Math.sin(30 * Math.PI / 180). One small conversion factor is the difference between −0.988 and 0.5.