What is the formula to convert degrees to radians?

Multiply the degree value by π/180: radians = degrees × π/180. For example, 120° = 120 × π/180 = 2π/3 ≈ 2.094 radians.

How do I convert radians back to degrees?

Multiply the radian value by 180/π: degrees = radians × 180/π. For example, π/4 radians × 180/π = 45°, and 1 radian ≈ 57.296°.

How many radians are in a full circle?

A full circle is 2π radians, roughly 6.283. A half circle is π radians (180°) and a quarter circle is π/2 radians (90°).

Why do mathematicians prefer radians?

Radians tie an angle directly to arc length on a unit circle, which makes calculus formulas simple. The derivative of sin(x) is cos(x) only in radians, and arc length equals radius times angle with no extra factor.

Degrees to Radians: The Conversion Explained

Published 3/4/2026 · 3 min read · Everyday calculators

Lena Hoffmann — Science & education writer at OneKitly

Mathematics · Physics

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In short

To convert an angle from degrees to radians, multiply by π/180: radians = degrees × π/180. To go back, multiply by 180/π. This works because a full turn is 360° and also 2π radians, so 180° equals exactly π radians. For example, 90° = 90 × π/180 = π/2 ≈ 1.571 radians, and 60° = π/3 ≈ 1.047 radians. Radians measure an angle by the arc length it cuts on a unit circle, which is why they are the natural unit in calculus and trigonometry.

Convert between degrees and radians with radians = degrees × π/180 (and back with × 180/π), understand why 180° = π, and learn the common angles by heart.

Why 180 degrees equals π radians

A radian is defined by the circle itself: it is the angle you get when the arc length equals the radius. Because the full circumference of a circle is 2π times its radius, a full turn is 2π radians. The same full turn is 360° by convention, so 360° = 2π radians, which halves neatly to 180° = π radians.

From 180° = π radians the two conversion factors fall out immediately. Dividing both sides by 180 gives 1° = π/180 radians, so you multiply degrees by π/180 to get radians. Dividing instead by π gives 1 radian = 180/π degrees, so you multiply radians by 180/π to get degrees. Numerically π/180 ≈ 0.01745 and 180/π ≈ 57.296.

The common angles worth memorizing

A handful of angles appear constantly, and knowing their radian forms saves time. 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, and 360° = 2π. Notice the pattern: each is a simple fraction of π, and the denominator tells you how many equal slices of a half-turn the angle represents.

To check one by hand, convert 45°: 45 × π/180 = 45π/180 = π/4, since 45 divides into 180 four times. Going the other way, π/3 radians is (π/3) × 180/π = 180/3 = 60°, because the π cancels. This cancellation is the reason radian answers are usually left as neat multiples of π rather than long decimals.

When each unit is the right choice

Degrees are convenient for everyday geometry, navigation, and construction because 360 divides evenly into many whole numbers. Radians are preferred in higher mathematics because they make formulas cleaner: the derivative of sin(x) is cos(x) only when x is in radians, and arc length is simply radius × angle when the angle is in radians.

A practical warning: calculators and programming languages default to radians in their trigonometric functions. If a spreadsheet or code returns a strange value for a sine you expected, the usual cause is a degree value fed into a function expecting radians. Convert first with × π/180, or use the dedicated degree functions where they exist.

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Frequently asked questions

What is the formula to convert degrees to radians?: Multiply the degree value by π/180: radians = degrees × π/180. For example, 120° = 120 × π/180 = 2π/3 ≈ 2.094 radians.
How do I convert radians back to degrees?: Multiply the radian value by 180/π: degrees = radians × 180/π. For example, π/4 radians × 180/π = 45°, and 1 radian ≈ 57.296°.
How many radians are in a full circle?: A full circle is 2π radians, roughly 6.283. A half circle is π radians (180°) and a quarter circle is π/2 radians (90°).
Why do mathematicians prefer radians?: Radians tie an angle directly to arc length on a unit circle, which makes calculus formulas simple. The derivative of sin(x) is cos(x) only in radians, and arc length equals radius times angle with no extra factor.

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