Areas Related to Circles
Board: GSEB | Class: Std 10
Comprehensive study notes for Areas Related to Circles by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.
Key Concepts
Circumference and Area
Circumference = 2πr. Area of circle = πr². These are the foundation formulas for all circle-related area calculations.
Area of a Sector
Sector is the region between two radii and the arc. Area = (θ/360) × πr² where θ is the sector angle. Length of arc = (θ/360) × 2πr.
Area of a Segment
Segment is the region between a chord and its arc. Area of minor segment = Area of sector – Area of triangle. Area of major segment = πr² – Area of minor segment.
Combined Figures
Problems involving combinations of circles, semicircles, squares, rectangles, triangles. Find the shaded area by adding/subtracting known areas.
Area of Annulus (Ring)
Area between two concentric circles = π(R² – r²) where R = outer radius, r = inner radius.
Important Formulas
| Circumference | C = 2πr |
| Area of Circle | A = πr² |
| Sector Area | A = (θ/360)πr² |
| Arc Length | L = (θ/360)2πr |
| Ring Area | A = π(R² - r²) |
Solved Examples
Example 1: Find the area of a sector of 60° in a circle of radius 14 cm.
Solution: Area = (60/360) × (22/7) × 14 × 14 = (1/6) × 616 = 102.67 cm².
Example 2: Find the area of minor segment of a circle of radius 14 cm with chord subtending 60° at center.
Solution: Sector area = 102.67 cm². Equilateral triangle area = (√3/4)(14)² = 84.87 cm². Segment = 102.67 – 84.87 = 17.8 cm².
Example 3: Find the area of a ring with outer radius 7 cm and inner radius 3 cm.
Solution: Area = π(49 – 9) = (22/7)(40) = 125.71 cm².
Practice Questions
- Find the area of a quadrant of a circle with radius 21 cm.
- The circumference of a circle is 88 cm. Find its area.
- Find the area of the shaded region formed by a square of side 14 cm with four semicircles on its sides.
- A chord subtends 90° at the center of a circle of radius 10 cm. Find area of segment.
- Find the area swept by minute hand of length 7 cm in 30 minutes.
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