Chapter 11 · Question 2

Derive the formulas for (i) the area of a sector of angle $\theta$ and radius $r$ , and (ii) the length of the arc of a sector of angle $\theta$ and radius $r$ .

Answer Revealed

Direct Answer:

Using the unitary method: For a full circle (angle

360^\circ

): area

= \pi r^2

, circumference

= 2\pi r

. For angle

1^\circ

: area of sector

= \frac{\pi r^2}{360}

, arc length

= \frac{2\pi r}{360}

. For angle

\theta

: (i)

\text{Area of sector} = \frac{\theta}{360} \times \pi r^2

, (ii)

\text{Arc length} = \frac{\theta}{360} \times 2\pi r

Simple Explanation

A full circle covers

360^\circ

. If you want just a part corresponding to angle

\theta

, you take that fraction of the whole. So area of sector =

(\theta/360) \times

(area of full circle), and arc length =

(\theta/360) \times

(full circumference).

Exam-Ready Structure

(i) Area of a sector of angle $\theta$ : The area of the entire circular region (full circle) is

\pi r^2

, which corresponds to a sector of angle

360^\circ

. By the unitary method: For angle

360^\circ

, area of sector

= \pi r^2

. For angle

1^\circ

, area of sector

= \frac{\pi r^2}{360}

. For angle

\theta

, area of sector

= \frac{\theta}{360} \times \pi r^2

. (ii) Length of an arc of a sector of angle $\theta$ : Similarly, the circumference (length of the entire boundary) of the circle is

2\pi r

, corresponding to angle

360^\circ

. For angle

1^\circ

, arc length

= \frac{2\pi r}{360}

. For angle

\theta

, arc length

= \frac{\theta}{360} \times 2\pi r

. Note: These formulas apply only when

\theta

is in degrees. The angle of the major sector is

(360^\circ - \theta)

, so its area

= \frac{360 - \theta}{360} \times \pi r^2

Key Points

Full circle area = πr² and circumference = 2πr (for 360°)
Area of sector = (θ/360) × πr²
Arc length = (θ/360) × 2πr
Derived using the unitary method (proportional to angle)
Major sector area = πr² − (minor sector area)

Derive the formulas for (i) the area of a sector of angle $\theta$ and radius $r$ , and (ii) the length of the arc of a sector of angle $\theta$ and radius $r$ .