Radian
A radian is the SI unit of angular measure defined as the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. Since the full circumference is 2πr, one complete revolution equals 2π radians, giving the exact conversion 180° = π radians. Radians are the natural unit for trigonometry and calculus because they make derivative formulas for trigonometric functions simple (d/dθ sin θ = cos θ holds only when θ is in radians).
Key Formula
θ (rad) = arc length s / radius r
LaTeX: \theta \text{ (rad)} = \frac{s}{r}
| Symbol | Meaning | Unit |
|---|---|---|
| θ | angle in radians | radians (rad) |
| s | arc length subtended by the angle | metres (m) |
| r | radius of the circle | metres (m) |
Worked Example
Problem
A wheel of radius 0.25 m rotates through an angle of 4 radians. What arc length does a point on the rim travel?
Solution
Step 1: Use the arc length formula s = rθ. Step 2: s = 0.25 m × 4 rad = 1.00 m. Step 3: As a check, 4 rad ≈ 4 × (180°/π) ≈ 229.2°, slightly more than half a revolution, which is geometrically reasonable for the arc being longer than the diameter (0.5 m).
Answer
The point on the rim travels an arc length of 1.00 m.
Degree–Radian Conversion for Common Angles
| Degrees | Radians (exact) | Radians (decimal) | Fraction of full turn |
|---|---|---|---|
| 30° | π/6 | 0.5236 | 1/12 |
| 45° | π/4 | 0.7854 | 1/8 |
| 60° | π/3 | 1.0472 | 1/6 |
| 90° | π/2 | 1.5708 | 1/4 |
| 180° | π | 3.1416 | 1/2 |
| 360° | 2π | 6.2832 | 1 |
Interactive Tools
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Related Terms
Sine Function
The sine function is a fundamental trigonometric function defined for an angle θ in a right triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse, extended to all real numbers via the unit circle. It is a periodic function with period 2π, amplitude 1, and range [−1, 1], producing a smooth oscillating wave. Sine is essential in modelling wave phenomena including sound, light, alternating current, and simple harmonic motion.
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system in which each point in the plane is specified by a radial distance r from a fixed origin (pole) and an angle θ measured from a fixed reference direction (polar axis), written as the ordered pair (r, θ). Unlike Cartesian coordinates that use perpendicular axes, polar coordinates are natural for describing curves with rotational symmetry such as circles, spirals, roses, and limaçons. They are widely used in physics (orbital mechanics, wave interference), engineering (antenna patterns), and complex number representation.
Tangent Function
The tangent function is defined as the ratio of the sine to the cosine of an angle (tan θ = sin θ / cos θ), or equivalently as the ratio of the opposite side to the adjacent side in a right triangle. Unlike sine and cosine, the tangent function has a period of π and is undefined at θ = π/2 + nπ (where n is any integer) because cosine equals zero at those points, producing vertical asymptotes. Tangent is fundamental in calculating slopes of lines, angles of elevation and depression, and in integral calculus substitutions.
The term "radian" was coined by Irish mathematician James Thomson (brother of Lord Kelvin) in an 1873 examination paper at Queen's College Belfast, derived from the Latin radius ("ray, spoke of a wheel"). The concept had been used implicitly by Roger Cotes (1682–1716), who recognised the natural relationship between arc length, radius, and angle.