Circular measure: radian measure, arc length, area of sector, small angle approximations

1. Circular Measure: Radian Measure

Imagine walking around a circular track. The distance you walk compared to the track’s radius tells you how many “radians” you’ve covered.

Definition: If a point moves along a circle of radius $r$ and sweeps out an arc of length $s$, the angle in radians is $$\theta=\frac{s}{r}.$$ The unit “radian” is dimensionless because it’s a ratio of two lengths.

Example: • 1 radian ≈ 57.3° (since 2π radians = 360°). • If you walk 10 m along a circle of radius 5 m, the angle is $$\theta=\frac{10}{5}=2\ \text{radians} \approx 114.6^\circ.$$

2. Arc Length

The arc length is the distance along the circle that a given angle covers.

Formula: $$s = r\,\theta,$$ where $s$ is the arc length, $r$ the radius, and $\theta$ the angle in radians.

Example: Radius $r = 5\text{ cm}$, angle $\theta = \frac{\pi}{3}$ rad. $$s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3}\ \text{cm} \approx 5.24\text{ cm}.$$

3. Area of a Sector

A sector is the “pie‑slice” part of a circle. Its area depends on the radius and the angle.

Formula: $$A = \tfrac{1}{2}\,r^2\,\theta.$$

Example: Radius $r = 4\text{ m}$, angle $\theta = \frac{\pi}{2}$ rad. $$A = \tfrac{1}{2}\times 4^2 \times \tfrac{\pi}{2} = 4\pi\ \text{m}^2 \approx 12.57\text{ m}^2.$$

4. Small Angle Approximations

When the angle $\theta$ is small (close to 0) and measured in radians, trigonometric functions can be approximated by simple expressions:

• sin θ ≈ θ (error < 0.1% for |θ| < 0.1 rad)
• tan θ ≈ θ (same accuracy)
• cos θ ≈ 1 – θ²⁄2

Example: θ = 0.1 rad
sin θ ≈ 0.1 (actual = 0.099833)
tan θ ≈ 0.1 (actual = 0.100334)
cos θ ≈ 1 – 0.1²⁄2 = 0.995 (actual = 0.995004)

Summary Table of Key Formulas