Areas Related to Circles

A circle is the set of all points in a plane at a fixed distance called the radius from a fixed point called the center. Understanding areas related to circles involves concepts such as sectors, segments, arcs, and annuli.

Concept Explanation

Circle: A circle is defined by its center \(O\) and radius \(r\). The circumference is the distance around the circle.

Radius: A line segment from the center to any point on the circumference.

Diameter: A chord passing through the center; longest chord of the circle, length \(d = 2r\).

Arc: A part of the circumference between two points.

Minor Arc: Arc length less than half the circumference.

Major Arc: Arc length greater than half the circumference.

Sector: Region bounded by two radii and the arc between them.

Segment: Region bounded by a chord and the arc it subtends.

Annulus: Region between two concentric circles with radii \(R\) and \(r\), where \(R > r\).

Formula Derivation

Let \(\theta\) be the central angle in degrees.

• Circumference: \[ C = 2\pi r \]
• Length of arc: \[ l = \frac{\pi r \theta}{180} \]
• Area of circle: \[ A = \pi r^2 \]
• Area of sector: \[ A_{sector} = \frac{\pi r^2 \theta}{360} \]
• Area of segment: \[ A_{segment} = A_{sector} - A_{triangle} \]
• Area of triangle formed by two radii and chord:
  \[ A_{triangle} = \frac{1}{2} r^2 \sin \theta \]
• Area of segment:
  \[ A_{segment} = \frac{\pi r^2 \theta}{360} - \frac{1}{2} r^2 \sin \theta \]
• Area of annulus:
  \[ A_{annulus} = \pi (R^2 - r^2) \]
• Perimeter of sector:
  \[ P_{sector} = 2r + l = 2r + \frac{\pi r \theta}{180} \]

Worked Illustrations

Example 1: Find the area of a sector with radius 4 cm and central angle 30°.

Given: \(r=4\) cm, \(\theta=30^\circ\)

Area of sector:

\[ A = \frac{\pi r^2 \theta}{360} = \frac{3.14 \times 4^2 \times 30}{360} = \frac{3.14 \times 16 \times 30}{360} = 4.19 \text{ cm}^2 \]

Area of corresponding major sector:

\[ A_{major} = \pi r^2 - A = 3.14 \times 16 - 4.19 = 50.24 - 4.19 = 46.05 \text{ cm}^2 \]

Example 2: Find the area of segment AYB where radius \(r=21\) cm and central angle \(\theta=120^\circ\).

Area of sector:

\[ A_{sector} = \frac{\pi r^2 \theta}{360} = \frac{22}{7} \times 21^2 \times \frac{120}{360} = \frac{22}{7} \times 441 \times \frac{1}{3} = 462 \text{ cm}^2 \]

Area of triangle formed by radii and chord:

\[ A_{triangle} = \frac{1}{2} r^2 \sin \theta = \frac{1}{2} \times 441 \times \sin 120^\circ = 220.5 \times \frac{\sqrt{3}}{2} = 190.95 \text{ cm}^2 \]

Area of segment:

\[ A_{segment} = A_{sector} - A_{triangle} = 462 - 190.95 = 271.05 \text{ cm}^2 \]

Practice Set

Level 1 – Easy

• Find the circumference of a circle with radius 7 cm.
• Calculate the area of a sector with radius 5 cm and central angle 60°.
• Find the length of an arc subtending 90° in a circle of radius 10 cm.

Level 2 – Moderate

• Calculate the area of a segment with radius 14 cm and central angle 90°.
• Find the area of an annulus with outer radius 10 cm and inner radius 6 cm.
• A wheel of radius 0.5 m makes 100 revolutions. Find the distance covered by the wheel.

Level 3 – Challenging

• Prove that the angle subtended by a diameter at the circumference is a right angle.
• Find the area of a segment where the chord subtends an angle of 60° at the center and radius is 12 cm.
• A sector has an area of 50 cm² and radius 10 cm. Find the central angle of the sector.

Answer Key

Level 1

• Circumference = \(2\pi r = 2 \times 3.14 \times 7 = 43.96\) cm
• Area of sector = \(\frac{\pi r^2 \theta}{360} = \frac{3.14 \times 25 \times 60}{360} = 13.09\) cm²
• Arc length = \(\frac{\pi r \theta}{180} = \frac{3.14 \times 10 \times 90}{180} = 15.7\) cm

Level 2

• Area of segment = \(\frac{\pi r^2 \theta}{360} - \frac{1}{2} r^2 \sin \theta = \frac{3.14 \times 196 \times 90}{360} - \frac{1}{2} \times 196 \times 1 = 153.86 - 98 = 55.86\) cm²
• Area of annulus = \(\pi (R^2 - r^2) = 3.14 (100 - 36) = 3.14 \times 64 = 200.96\) cm²
• Distance covered = circumference \(= 2\pi r = 2 \times 3.14 \times 0.5 = 3.14\) m; Total distance = \(3.14 \times 100 = 314\) m

Level 3

• Proof: The angle subtended by a diameter at the circumference is a right angle because the triangle formed is right angled at the circumference point (Thales' theorem).
• Area of segment = \(\left( \frac{\pi}{6} - \frac{\sqrt{3}}{4} \right) r^2 = \left( 0.5236 - 0.4330 \right) \times 144 = 12.7\) cm²
• Area of sector = 50 cm², radius = 10 cm
  \[ 50 = \frac{\pi \times 10^2 \times \theta}{360} \Rightarrow \theta = \frac{50 \times 360}{3.14 \times 100} = 57.32^\circ \]

Quick Reference

Glossary

• Radius: Distance from center to circumference.
• Diameter: Longest chord passing through center.
• Chord: Line segment joining two points on circumference.
• Arc: Part of circumference between two points.
• Sector: Region bounded by two radii and arc.
• Segment: Region bounded by chord and arc.
• Annulus: Ring-shaped region between two concentric circles.
• Central Angle: Angle subtended at center by two radii.
• Minor Arc: Smaller arc between two points.
• Major Arc: Larger arc between two points.