Areas Related to Circles (Class 10): Notes और MCQ Practice

आसान भाषा में

यह chapter circle के different parts जैसे circumference, area, sector और segment के measurements के बारे में है। सोचो एक गोल pizza है: उसका crust (boundary) circumference है, पूरा pizza circle का area है, एक slice sector है, और chord से कटा हुआ किनारा segment है! बस इन्हीं के formulas से questions solve करने हैं।

Important Formulas & Points

• Circumference of circle = 2 × π × r
• Area of circle = π × r²
• Length of an Arc (angle θ) = (θ ÷ 360) × 2 × π × r
• Area of a Sector (angle θ) = (θ ÷ 360) × π × r²
• Area of Minor Segment = Area of Sector − Area of Triangle
• Area of Major Segment = Area of Circle − Area of Minor Segment
• Area of Major Sector = Area of Circle − Area of Minor Sector (या angle 360 − θ use करें)

2 Memory Tricks

1. Arc vs Sector Formula: Arc सिर्फ boundary (perimeter) का हिस्सा है, इसलिए इसमें Circumference आता है: (θ ÷ 360) × 2πr. Sector पूरे flat surface (area) का हिस्सा है, इसलिए इसमें Area आता है: (θ ÷ 360) × πr².
2. Pizza Slice Trick: 'Sector' = Slice (दो radii और arc से घिरा). 'Segment' = Straight knife cut (chord और arc के बीच का हिस्सा).

2 Common Board-Exam Mistakes

1. Major vs Minor miss करना: Question में अगर 'Major' sector या segment पूछा है, तो student अक्सर minor निकाल कर छोड़ देते हैं। Total area में से minor subtract करना न भूलें!
2. π की wrong value: हमेशा question के end में देखें—अगर π = 3.14 दिया है तो वही लें, नहीं तो 22/7 लें। गलत value लेने से final answer में marks कट सकते हैं.

Class 10 Mathematics: Areas Related to Circles

1. Chapter in Short

In everyday life, we see many circular shapes—car wheels, clocks, circular fields, and round pizzas. In this chapter, we build upon the basic concepts of the perimeter (circumference) and area of a circle. We learn how to calculate:

• The length of a specific portion of the circular boundary (arc length).
• The area of a slice of a circle bounded by two radii and an arc (sector).
• The area of a region bounded by a chord and an arc (segment).

2. Key Concepts Explained Simply

A. Perimeter and Area of a Circle

• Circumference: Think of a round running track. If you walk all the way around the track once and return to your starting point, the total distance you covered is the perimeter or circumference.
• Area: The total ground covered inside that round running track is the area of the circle.

B. Sector of a Circle

• Imagine a whole round pizza. When you take out a slice cut from the center to the edge, that slice is a sector.
• The boundary of a sector consists of two radii and the curved circular edge between them.
• Minor Sector: The smaller slice (angle at center θ < 180°).
• Major Sector: The remaining bigger portion of the pizza (angle at center = 360° − θ).

C. Arc of a Circle

• The curved crust on the outer edge of your pizza slice is an arc.
• Its length depends on the angle made at the center of the circle.

D. Segment of a Circle

• Imagine cutting across a circular cookie with a straight knife cut (a chord) that does not pass through the center.
• The piece cut off between the straight line (chord) and the curved edge is called a segment.
• Minor Segment: The smaller region between the chord and the smaller arc.
• Major Segment: The larger region of the circle left behind.

3. Important Definitions, Formulas, and Facts

Basic Circle Formulas

• Radius ($r$): Distance from the center to any point on the boundary.
• Diameter ($d$): $d = 2 × r$
• Circumference ($C$): $C = 2 × π × r$ or $C = π × d$
• Area of a Circle ($A$): $A = π × r²$
• Value of π: Take $π = 22 ÷ 7$ unless explicitly given as $3.14$.

Sector and Arc Formulas

For a circle of radius $r$ and a sector with central angle $θ$ (in degrees):

• Length of an Arc ($l$):

$l = (θ ÷ 360) × 2 × π × r$

• Area of a Sector:

$Area = (θ ÷ 360) × π × r²$

(Alternatively, Area of sector = 1 ÷ 2 × arc length × radius = 1 ÷ 2 × l × r)

• Perimeter of a Sector:

$Perimeter = Length of arc + 2 × r = l + 2r$

• Area of Major Sector:

$Area = π × r² − Area of minor sector$

Segment Formulas

• Area of Minor Segment:

$Area = Area of corresponding sector − Area of triangle formed by the two radii and the chord$

• Area of Major Segment:

$Area = Area of circle − Area of minor segment$

Useful Area of Triangle Formulas for Central Angle $θ$:

• If $θ = 90°$: Triangle is a right-angled isosceles triangle.

$Area = 1 ÷ 2 × base × height = 1 ÷ 2 × r × r = 1 ÷ 2 × r²$

• If $θ = 60°$: Triangle formed by two radii is an equilateral triangle.

$Area = (√3 ÷ 4) × r²$

4. Solved Examples

Example 1: Basic Sector Area

Question: Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60°. (Use $π = 22 ÷ 7$)

Solution:

• Radius ($r$) = 6 cm
• Angle of the sector ($θ$) = 60°
• Area of sector = $(θ ÷ 360) × π × r²$
• Area = $(60 ÷ 360) × (22 ÷ 7) × 6 × 6$
• Area = $(1 ÷ 6) × (22 ÷ 7) × 36$
• Area = $(22 ÷ 7) × 6 = 132 ÷ 7$ cm² (or approx. 18.86 cm²)

Example 2: Clock Problem (Frequently asked in Board Exams)

Question: The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. (Use $π = 22 ÷ 7$)

Solution:

• The minute hand completes a full round of 360° in 60 minutes.
• Angle swept in 1 minute = $360° ÷ 60 = 6°$
• Angle swept in 5 minutes ($θ$) = $5 × 6° = 30°$
• Length of the minute hand acts as radius ($r$) = 14 cm
• Area swept = Area of sector with $θ = 30°$

$Area = (θ ÷ 360) × π × r²$

$Area = (30 ÷ 360) × (22 ÷ 7) × 14 × 14$

$Area = (1 ÷ 12) × (22 ÷ 7) × 196$

$Area = (1 ÷ 12) × 22 × 28$

$Area = 616 ÷ 12 = 154 ÷ 3$ cm² (or approx. 51.33 cm²)

Example 3: Finding Area of a Segment (Right-angled Triangle)

Question: A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the corresponding minor segment. (Use $π = 3.14$)

Solution:

• Radius ($r$) = 10 cm
• Central angle ($θ$) = 90°
• Step 1: Find Area of Sector

$Area of sector = (90 ÷ 360) × 3.14 × 10²$

$= (1 ÷ 4) × 3.14 × 100$

$= (1 ÷ 4) × 314 = 78.5$ cm²

• Step 2: Find Area of the Triangle

Since $θ = 90°$, base = $r = 10$ cm and height = $r = 10$ cm.

$Area of triangle = 1 ÷ 2 × base × height = 1 ÷ 2 × 10 × 10 = 50$ cm²

• Step 3: Find Area of Minor Segment

$Area of minor segment = Area of sector − Area of triangle$

$= 78.5 − 50 = 28.5$ cm²

Example 4: Arc Length and Segment (Equilateral Triangle Case)

Question: In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. Find:

1. The length of the arc
2. The area of the sector formed by the arc

(Use $π = 22 ÷ 7$)

Solution:

• Radius ($r$) = 21 cm, Angle ($θ$) = 60°
• Part 1: Length of the arc ($l$)

$l = (θ ÷ 360) × 2 × π × r$

$l = (60 ÷ 360) × 2 × (22 ÷ 7) × 21$

$l = (1 ÷ 6) × 2 × 22 × 3$

$l = (1 ÷ 6) × 132 = 22$ cm

• Part 2: Area of the sector

$Area = (θ ÷ 360) × π × r²$

$Area = (60 ÷ 360) × (22 ÷ 7) × 21 × 21$

$Area = (1 ÷ 6) × 22 × 3 × 21$

$Area = (1 ÷ 6) × 1386 = 231$ cm²

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid:

1. Confusing Sector with Segment:

• A sector has two straight sides going to the center (radius + radius) and one curved edge.
• A segment is bounded by a straight line cut across (chord) and a curved edge.

1. Perimeter of a Sector:

• Students often calculate only the arc length. Remember that the perimeter of a sector includes the two radii: $Perimeter = Arc\ length + 2r$.

1. Using the Wrong Value of π:

• Always check the question paper carefully. If it mentions "Use $π = 3.14
  quot;, do not use $22 ÷ 7$, otherwise your decimal answer may vary slightly.

1. Forgetting Proper Units:

• Arc length and perimeter are lengths, so units are cm or m.
• Areas are surface measurements, so units must be cm² or m².

Quick Revision Points:

• Area of circle = $π × r²$
• Circumference = $2 × π × r$
• Arc length = $(θ ÷ 360) × 2 × π × r$
• Sector area = $(θ ÷ 360) × π × r²$
• Minor segment area = Sector area − Area of triangle
• Major segment area = Area of circle − Minor segment area