Areas Related to Circles: Class 10 Maths NCERT Guide

Welcome, student! Have you ever wondered how to calculate the area of a pizza slice, or the distance covered by a wheel in one rotation? This chapter, 'Areas Related to Circles', moves beyond the simple area and circumference formulas you learned in earlier classes. Here, we'll dive into the fascinating parts of a circle, like sectors (the pizza slice) and segments (the part cut off by a chord). You'll master the formulas to calculate their areas and the lengths of their arcs. This chapter is a beautiful blend of geometry and arithmetic, crucial for scoring well in your CBSE board exams. By the end of this guide, you will be able to confidently solve problems involving sectors, segments, and even complex shapes made by combining circles with other figures like squares and triangles. Let's begin this exciting journey!

Key Terms and Formulas

Circumference: The distance around the circle. Formula: C = 2πr, where 'r' is the radius.
Area of a Circle: The space enclosed by the circle. Formula: A = πr², where 'r' is the radius.
Sector: The region enclosed by two radii and the corresponding arc. It looks like a slice of pizza. The smaller area is the minor sector and the rest is the major sector.
Segment: The region enclosed by a chord and the corresponding arc. The smaller area is the minor segment and the larger area is the major segment.
Length of an Arc of a Sector: The length of the curved part of a sector. Formula: (θ/360°) × 2πr, where θ is the angle of the sector in degrees.
Area of a Sector: The area of a pizza-slice shape. Formula: (θ/360°) × πr², where θ is the angle of the sector in degrees.

How to Calculate the Area of a Segment

Finding the area of a segment might seem tricky, but it's a simple two-step process. A segment is the area enclosed by a chord and an arc. To find its area, we first need to visualize the sector that contains this segment. The area of the segment is what's left over when you take the area of the corresponding sector and subtract the area of the triangle formed by the two radii and the chord.

So, the master formula is:

Area of Segment = Area of the corresponding Sector – Area of the corresponding Triangle.

Let's say the sector has an angle θ and radius r. The area of the sector is (θ/360°) × πr². The triangle inside this sector is formed by the two radii and the chord. You can find its area using formulas like ½ × base × height, or if you know trigonometry, ½ r² sin(θ). Once you have both these areas, just subtract the triangle's area from the sector's area. That's it! You've successfully found the area of the segment.

Solved Example: Finding Sector and Segment Area

Question: A chord of a circle with a radius of 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle. (Use π = 22/7). Solution: Given: Radius (r) = 14 cm Angle at the centre (θ) = 60° Step 1: Calculate the Area of the Minor Sector (OAPB) The formula for the area of a sector is (θ/360°) × πr². Area of Sector = (60°/360°) × (22/7) × (14)² Area of Sector = (1/6) × (22/7) × 14 × 14 Area of Sector = (1/6) × 22 × 2 × 14 = (616/6) = 308/3 cm². Area of Sector ≈ 102.67 cm². Step 2: Calculate the Area of the Triangle (ΔOAB) Since the two sides OA and OB are radii (14 cm) and the angle between them is 60°, ΔOAB is an isosceles triangle. Because the angle at O is 60°, the other two angles (∠OAB and ∠OBA) are also 60°. Therefore, ΔOAB is an equilateral triangle with side length 14 cm. The formula for the area of an equilateral triangle is (√3/4) × (side)². Area of ΔOAB = (√3/4) × (14)² Area of ΔOAB = (√3/4) × 196 = 49√3 cm². Using √3 ≈ 1.73, Area of ΔOAB ≈ 49 × 1.73 = 84.77 cm². Step 3: Calculate the Area of the Minor Segment Area of Minor Segment = Area of Minor Sector – Area of ΔOAB Area of Minor Segment = (308/3) - 49√3 cm² Area of Minor Segment ≈ 102.67 - 84.77 = 17.9 cm². Final Answer: The area of the minor segment is approximately 17.9 cm².

Exam Tips & Common Mistakes to Avoid

To score full marks in this chapter, pay attention to these common pitfalls:

Sector vs. Segment: A very common mistake is to calculate the area of the sector when asked for the segment. Remember, for a segment, you MUST subtract the area of the triangle.
Value of π: Always check if the question specifies a value for π (like 3.14 or 22/7). If not specified, using 22/7 is generally a good practice unless the radius is a multiple of 10.
Units: Be careful with units! Area is always in square units (cm², m²) while length (circumference, arc length) is in linear units (cm, m). Write the correct units in your final answer.
Major vs. Minor: Read the question carefully to see if it asks for the 'major' or 'minor' sector/segment. To find the major sector/segment, you can calculate the minor part and subtract it from the total area of the circle.

Practice Questions with Solutions

Q: Find the area of a sector of a circle with a radius of 6 cm if the angle of the sector is 60°. A: Step 1: Identify the given values. Radius (r) = 6 cm and Angle (θ) = 60°. Step 2: Use the formula for the area of a sector: Area = (θ/360°) × πr². Step 3: Substitute the values: Area = (60/360) × (22/7) × 6 × 6 = (1/6) × (22/7) × 36. Step 4: Calculate the final answer: Area = (22/7) × 6 = 132/7 cm². Final answer: The area of the sector is 132/7 cm² or approximately 18.86 cm².
Q: Find the area of a quadrant of a circle whose circumference is 44 cm. A: Step 1: Find the radius from the circumference. C = 2πr => 44 = 2 × (22/7) × r => 44 = (44/7) × r => r = 7 cm. Step 2: A quadrant means the angle at the centre is 90°. So, θ = 90°. Step 3: Use the formula for the area of a sector (quadrant): Area = (θ/360°) × πr². Step 4: Substitute the values: Area = (90/360) × (22/7) × 7² = (1/4) × (22/7) × 49 = (1/4) × 22 × 7 = 154/4 = 77/2 cm². Final answer: The area of the quadrant is 38.5 cm².
Q: A horse is tied to a corner of a square field of side 20 m by a rope 14 m long. Find the area of the field the horse can graze. A: Step 1: The area the horse can graze is a sector of a circle. The rope length is the radius (r = 14 m). Step 2: Since the horse is tied at the corner of a square, the angle of the sector is 90° (θ = 90°). Step 3: Use the formula for the area of a sector: Area = (θ/360°) × πr². Step 4: Substitute the values: Area = (90/360) × (22/7) × 14² = (1/4) × (22/7) × 14 × 14 = (1/4) × 22 × 2 × 14 = 616/4 = 154 m². Final answer: The horse can graze an area of 154 m².
Q: An umbrella has 8 ribs that are equally spaced. Assuming the umbrella is a flat circle of radius 42 cm, find the area between two consecutive ribs. A: Step 1: The area between two consecutive ribs forms a sector. The total angle in a circle is 360°. Step 2: Since there are 8 equally spaced ribs, the angle of the sector between two ribs is θ = 360°/8 = 45°. Step 3: The radius of the circle is given as r = 42 cm. Step 4: Use the formula for the area of a sector: Area = (θ/360°) × πr² = (45/360) × (22/7) × 42² = (1/8) × (22/7) × 42 × 42 = (1/8) × 22 × 6 × 42 = 5544 / 8 = 693 cm². Final answer: The area between two consecutive ribs is 693 cm².

Frequently Asked Questions

What is the difference between a sector and a segment of a circle?

A sector is a region bounded by two radii and an arc, resembling a slice of pizza. A segment is a region bounded by a chord and an arc. Think of a sector as 'pizza-slice shaped' and a segment as the 'crust part' cut off by a straight line.

When should I use π = 22/7 versus π = 3.14?

Always follow the instructions in the question. If a value is specified, use that. If not, a good rule of thumb is to use π = 22/7 when the radius is a multiple of 7, as it simplifies calculations. Otherwise, π = 3.14 is a good choice.

How is the area of a major segment calculated?

The easiest way is to find the area of the entire circle (πr²) and then subtract the area of the corresponding minor segment. Alternatively, you can calculate the area of the major sector and add the area of the triangle.

What are some real-life applications of finding areas related to circles?

These concepts are used everywhere! They are used in designing parks and gardens (calculating lawn area), in engineering (designing gears and machine parts), in architecture (designing windows and domes), and even in cooking (dividing a pizza or cake equally).