A-Level — mathsStandard1 min read

Radians, Arc Length, and Sector Area

Work in radians at A-Level. Calculate arc lengths and sector areas using radian measure.

Tutor AI Team
Tutor AI Team

Radians are the natural unit for measuring angles in mathematics. All A-Level calculus involving trig functions requires radians.

Core Concepts

Radians

2π2\pi radians = 360°. π\pi radians = 180°.

Convert: degrees × π180\frac{\pi}{180} = radians. Radians × 180π\frac{180}{\pi} = degrees.

Arc Length

s=rθs = r\theta (where θ\theta in radians)

Sector Area

A=12r2θA = \frac{1}{2}r^2\theta

Segment Area

Asegment=12r2(θ−sin⁡θ)A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)

Worked Example: Example 1

Problem

Arc length: r=8r = 8, θ=π3\theta = \frac{\pi}{3}. s=8×π3=8π3s = 8 \times \frac{\pi}{3} = \frac{8\pi}{3}.

Solution

Worked Example: Example 2

Problem

Sector area: r=10r = 10, θ=1.2\theta = 1.2. A=12(100)(1.2)=60A = \frac{1}{2}(100)(1.2) = 60.

Solution

Worked Example: Example 3

Problem

Segment: r=6r = 6, θ=π4\theta = \frac{\pi}{4}. A=12(36)(π4−sin⁡π4)=18(π4−22)≈1.41A = \frac{1}{2}(36)(\frac{\pi}{4} - \sin\frac{\pi}{4}) = 18(\frac{\pi}{4} - \frac{\sqrt{2}}{2}) \approx 1.41.

Solution

Practice Problems

    1. Convert 150° to radians.
    1. Arc length: r=12r = 12, θ=0.8\theta = 0.8 rad.
    1. Segment area: r=5r = 5, θ=π3\theta = \frac{\pi}{3}.

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Key Takeaways

  • ✓

    s=rθs = r\theta. A=12r2θA = \frac{1}{2}r^2\theta. θ\theta must be in radians.

  • ✓

    Segment = sector − triangle: 12r2(θ−sin⁡θ)\frac{1}{2}r^2(\theta - \sin\theta).

Tags:
#a-level#maths#pure#trigonometry#radians

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