CBSE Class 10 Maths Chapter 12: Areas Related To Circles Notes

Key Concepts & Formulas

• Area of a Circle (A): πr², where 'r' is the radius.
• Circumference of a Circle (C): 2πr, where 'r' is the radius.
• Area of a Sector (A_sector): (θ/360°) × πr², where 'θ' is the central angle in degrees and 'r' is the radius.
• Length of an Arc of a Sector (L_arc): (θ/360°) × 2πr, where 'θ' is the central angle in degrees and 'r' is the radius.
• Area of a Minor Segment (A_minor_seg): Area of corresponding Sector - Area of ΔOAB. (ΔOAB is formed by radii OA, OB and chord AB). Formula for ΔOAB is (1/2)r²sinθ.
• Area of a Major Segment (A_major_seg): Area of Circle - Area of Minor Segment.
• Area of a Quadrant: (1/4) × πr² (since angle is 90°).
• Value of π: Generally taken as 22/7 or 3.14, as specified in the problem. If not specified, 22/7 is often preferred.

Key Terms Defined

Circle

A closed plane figure whose boundary (circumference) consists of all points equidistant from a fixed point (center).

Radius (r)

The distance from the center of the circle to any point on its circumference.

Diameter (d)

A straight line segment that passes through the center of the circle and has its endpoints on the circumference. d = 2r.

Circumference

The total distance around the circle; essentially, its perimeter.

Arc

A continuous portion of the circumference of a circle.

Sector

The region of a circle enclosed by two radii and the arc connecting their endpoints. Think of it as a 'slice of pizza'.

Segment

The region of a circle enclosed by a chord and an arc. It is the area left when a triangle is removed from a sector.

Quadrant

A sector of a circle formed by two perpendicular radii, enclosing a central angle of 90 degrees. It is one-fourth of a circle.

Understanding Areas of Sectors and Segments

Mastering the calculation of areas related to sectors and segments is fundamental for this chapter. A sector of a circle is essentially a fraction of the entire circle's area, determined by the central angle (θ) it subtends. If the central angle is 360°, it's the whole circle. If it's 180°, it's a semicircle. Therefore, the area of a sector is proportional to the ratio of its central angle to 360°, multiplied by the total area of the circle (πr²). The formula, (θ/360°) × πr², directly reflects this proportionality. Similarly, the length of an arc of a sector is proportional to the same ratio, applied to the circle's circumference, giving (θ/360°) × 2πr.

Calculating the area of a segment requires a slightly more involved approach. A segment is the region bounded by a chord and an arc. To find its area, we first consider the corresponding sector. The area of the segment is then obtained by subtracting the area of the triangle formed by the two radii and the chord from the area of that sector. For a minor segment, this subtraction yields the desired area. Specifically, Area of Minor Segment = Area of Sector - Area of ΔOAB. The area of the triangle ΔOAB (where O is the center and A, B are points on the circle) can be calculated using the formula (1/2)r²sinθ. For common angles: for θ = 90°, Area of ΔOAB = (1/2)r²; for θ = 60°, ΔOAB is equilateral, so Area of ΔOAB = (√3/4)r². If you need to find the Area of a Major Segment, simply subtract the area of the minor segment from the total area of the circle. Always ensure your central angle is in degrees for these formulas.

Worked Examples

• Example 1: Area of a Sector Q: Find the area of a sector of a circle with radius 14 cm and a central angle of 45°. (Use π = 22/7) A: Given, r = 14 cm, θ = 45°. Area of sector = (θ/360°) × πr² = (45/360) × (22/7) × 14² = (1/8) × (22/7) × 196 = (1/8) × 22 × 28 = 1 × 11 × 7 = 77 cm².
• Example 2: Area of a Minor Segment Q: A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the minor segment. (Use π = 3.14) A: Given, r = 10 cm, θ = 90°. Area of sector = (90/360) × πr² = (1/4) × 3.14 × 10² = (1/4) × 3.14 × 100 = 314/4 = 78.5 cm². Area of triangle OAB = (1/2) × base × height (since θ = 90°, it's a right-angled triangle) = (1/2) × r × r = (1/2) × 10 × 10 = 50 cm². Area of minor segment = Area of sector - Area of ΔOAB = 78.5 - 50 = 28.5 cm².

Comparison: Circle, Sector & Segment Formulas

Exam Strategy & Common Pitfalls

To score well in this chapter, consistent practice and attention to detail are key:

• Read Carefully: Distinguish between questions asking for arc length, area of sector, area of segment, or perimeter of sector/segment.
• Value of π: Always use the value of π specified in the question (e.g., 22/7 or 3.14). If not specified, 22/7 is generally a safe bet.
• Units: Never forget to write the correct units for your answers. Lengths are in cm/m, and areas are in cm²/m².
• Diagrams: For complex problems involving combinations of figures (e.g., area of a square with inscribed circles), draw clear diagrams to visualize the problem and break it down into simpler shapes.
• Angle in Degrees: Ensure the central angle (θ) is always in degrees when using the (θ/360°) factor in the formulas.
• Triangle Area: Be proficient in calculating the area of the triangle formed by the two radii and the chord, as it's crucial for segment area calculations. Remember (1/2)r²sinθ, or (1/2) base height for right-angled triangles.

Quick Revision Check

• Q: What is the formula for the perimeter of a semicircle? A: πr + 2r (arc length + diameter).
• Q: If the area of a circle is 154 cm², what is its radius? A: πr² = 154 => (22/7)r² = 154 => r² = 154 (7/22) = 7 7 = 49 => r = 7 cm.
• Q: How do you find the area of a major sector when the minor sector's angle is given as θ? A: Area of Major Sector = Area of Circle - Area of Minor Sector, or directly using the angle (360° - θ) in the sector area formula: [(360° - θ)/360°] × πr².
• Q: A circular park has a radius of 21 m. A 3.5 m wide path runs along its boundary inside. What is the radius of the inner circular area? A: Radius of inner circular area = Outer radius - width of path = 21 m - 3.5 m = 17.5 m.

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