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Radian — Definition, Formula & Examples

Radian

A unit for measuring angles. 180° = π radians, and 360° = 2π radians. The number of radians in an angle equals the number of radii it takes to measure a circular arc described by that angle.

Note: 360° equals 2π radians because a complete circular arc has length equal to 2π times the radius.

A circle with central angle θ subtended by two radii r; arc length s highlighted terracotta. θ = s/r. 180° = π radians, 360° = 2π radians.

Commonly used angles in degrees and radians

See also

Special angles

Key Formula

θ=sr\theta = \frac{s}{r}
Where:
  • θ\theta = Measure of the central angle in radians
  • ss = Arc length (the distance along the curved part of the circle)
  • rr = Radius of the circle

Worked Example

Problem: A circle has a radius of 6 cm. A central angle cuts off an arc that is 9 cm long. Find the measure of the angle in radians.
Step 1: Write down the known values.
s=9 cm,r=6 cms = 9 \text{ cm}, \quad r = 6 \text{ cm}
Step 2: Apply the radian formula.
θ=sr=96\theta = \frac{s}{r} = \frac{9}{6}
Step 3: Simplify the fraction.
θ=1.5 radians\theta = 1.5 \text{ radians}
Answer: The central angle measures 1.5 radians.

Another Example

This example shows how to convert from degrees to radians, rather than computing an angle from arc length and radius.

Problem: Convert 120° to radians.
Step 1: Use the conversion relationship: 180° equals π radians. This gives the conversion factor.
1°=π180 radians1° = \frac{\pi}{180} \text{ radians}
Step 2: Multiply the degree measure by the conversion factor.
120°×π180120° \times \frac{\pi}{180}
Step 3: Simplify by dividing both 120 and 180 by their greatest common factor, 60.
120π180=2π3\frac{120\pi}{180} = \frac{2\pi}{3}
Answer: 120° equals 2π/3 radians (approximately 2.094 radians).

Frequently Asked Questions

How do you convert degrees to radians?
Multiply the degree measure by π/180. For example, 90° × (π/180) = π/2 radians. This works because 180° and π radians represent the same angle, so π/180 is the exact conversion factor.
Why do we use radians instead of degrees?
Radians connect angles directly to arc length through the simple formula θ = s/r, with no extra conversion constants needed. In calculus and higher math, radians make formulas much cleaner — for instance, the derivative of sin(x) is cos(x) only when x is in radians. Degrees would require an extra factor of π/180 in every derivative and integral involving trig functions.
How many radians are in a full circle?
A full circle contains 2π radians, which is approximately 6.2832 radians. This follows from the circumference formula: a full circle's arc length is 2πr, so θ = 2πr / r = 2π.

Radians vs. Degrees

RadiansDegrees
DefinitionAngle measured by the ratio of arc length to radiusAngle measured as a fraction of a full rotation, divided into 360 equal parts
Full circle2π ≈ 6.283360°
Right angleπ/2 ≈ 1.57190°
Conversion formularadians = degrees × π/180degrees = radians × 180/π
When to useCalculus, physics, and any formula involving arc length or angular velocityEveryday measurement, geometry, navigation, and construction

Why It Matters

Radians appear throughout trigonometry, precalculus, and calculus courses. Nearly every formula involving angular velocity, arc length, or sector area uses radians. When you study calculus, radian measure is essential because the standard derivative rules for sine and cosine — and all trig-based formulas in physics — assume angles are in radians.

Common Mistakes

Mistake: Forgetting to set your calculator to radian mode when a problem uses radians.
Correction: Always check whether a problem gives angles in degrees or radians, then set your calculator accordingly. Computing sin(π) in degree mode gives sin(3.14°) ≈ 0.0548, not the correct answer of 0.
Mistake: Using the wrong conversion factor — multiplying by 180/π when converting degrees to radians (or vice versa).
Correction: Remember: degrees to radians, multiply by π/180 (you're making the number smaller, since π ≈ 3.14 is much less than 180). Radians to degrees, multiply by 180/π (you're making the number larger).

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