CH – 11 AREAS RELATED TO CIRCLES
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Master circle measurements step by step and build strong confidence to score top marks in your Class 10 Board Exam!
Chapter Overview
In this chapter, you will learn how to measure different parts and regions of a circle:
- Parts of a circle: Refreshing basic terms like centre, radius, diameter, chord, and circumference.
- Arc of a circle: Understanding the curved boundary of a circle.
- Sector of a circle: Finding the area of a slice of a circle bounded by two radii.
- Minor and major sector: Identifying small and large sectors based on central angles.
- Segment of a circle: Finding the region bounded by a chord and an arc.
- Minor and major segment: Distinguishing between small and large circular segments.
- Length of an arc: Calculating the curved length of a part of the circle boundary.
- Area of a sector: Calculating sector areas using central angles.
- Area of a segment: Subtracting triangle areas from sector areas.
- Problems involving sectors and segments: Applying formulas to practical daily life objects like clocks, umbrellas, and table covers.
🎯 Board Exam Focus
Practise the following core skills carefully to perform well in your exam:
- Understanding sectors and segments clearly from any given diagram.
- Identifying minor and major regions correctly.
- Finding the length of an arc using the radius and central angle.
- Finding the area of a sector accurately.
- Finding the area of a segment step by step.
- Using the correct formula for length (perimeter) versus area.
- Understanding which area needs to be added or subtracted in multi-part figures.
- Solving questions step by step with correct units.
Most Important Concepts
Quick Revision: Important Parts of a Circle
Before finding areas, let us revise the basic parts of a circle:
- Centre: The middle point inside the circle, usually named O.
- Radius (r): The straight line distance from the centre to any point on the circle boundary.
- Diameter (d): A straight line passing through the centre connecting two points on the circle. Diameter = 2 × radius.
- Chord: A straight line joining any two points on the circle boundary.
- Arc: A curved part of the outer boundary (circumference) of the circle.
Plaintext
.---. Legend / Labels:
/ | \ ----------------
/ | r \ O = Centre
| A--O--B | r = Radius (OC)
/ \ | / \ AB = Diameter (A to B through O)
D---E | / \ DE = Chord (joins two points on circle)
\ | / / ACB = Curved Arc
'-----C-----'
What is an Arc?
An arc is simply a curved portion of the circle boundary.
If you pick two points A and B on a circle, they divide the boundary into two curved paths:
- Minor Arc: The smaller curved path between points A and B.
- Major Arc: The larger curved path between points A and B.
Plaintext
.--- Q ---.
/ \ AQB = Major Arc (Larger curved path)
/ \ APB = Minor Arc (Smaller curved path)
| O |
\ / Points A and B divide the outer
\ / boundary into two arcs.
A -- P -- B
What is a Sector?
A sector is the region inside a circle enclosed by two radii and the corresponding arc.
- Minor Sector: The smaller region formed by two radii and the minor arc. The central angle θ is less than 180°.
- Major Sector: The larger region formed by two radii and the major arc. Its central angle is 360° − θ.
- Angle of the Sector (θ): The angle made at the centre O by the two radii.
Plaintext
.--- Q ---.
/ \ O = Centre
/ MAJOR \ OA, OB = Two Radii
| SECTOR | θ = Central Angle
\ O / OAPB = Minor Sector (Shaded)
\ /θ\ / OAQB = Major Sector (Unshaded)
'- A---B -'
P
🧠 Easy Way to Remember a Sector
Think of a sector as a slice of pizza or a slice of a birthday cake!
The point of the slice is at the centre O, and the two straight sides are the two radii.
What is a Segment?
A segment is the region inside a circle enclosed by a chord and the corresponding arc.
- Minor Segment: The smaller region between the chord and the minor arc.
- Major Segment: The larger region between the chord and the major arc.
Notice that a segment does NOT touch the centre O directly. It is formed by drawing a straight line (chord) across the circle.
Plaintext
.--- Q ---.
/ \ AB = Chord
/ MAJOR \ APB = Minor Arc
| SEGMENT |
\ O / Shaded APB = Minor Segment
\ / Unshaded AQB = Major Segment
A=========B
P
🧠 Sector vs Segment
Here is the simplest way to remember the difference:
- Sector: Enclosed by Two Radii + One Arc (Looks like a pizza slice).
- Segment: Enclosed by One Chord + One Arc (Looks like a cut cap or bow segment).
Plaintext
[ SECTOR ] [ SEGMENT ]
.---. .---.
/ \ / \
| O | | O |
\ / \ / \ /
'A---B' 'A===B'
(Two Radii + Arc) (One Chord + Arc)
📌 Length of an Arc
The full distance around a circle is its circumference:
Circumference = 2πr
A full circle has a central angle of 360°. If an arc subtends an angle of θ° at the centre, the arc length is just a fraction of the total circumference.
Arc Length Formula
Length of an arc = (θ / 360) × 2πr
Where:
- θ = angle subtended at the centre in degrees
- r = radius of the circle
- π = pi (usually 22/7 or 3.14)
Plaintext
.--- Q ---.
/ \ Radius = r
/ O \ Central Angle = θ
| /θ\ | Highlighted Arc APB = (θ/360) × 2πr
\ / r \ /
\ / \ / It is a fraction of the full
A ---P--- B circumference (2πr).
Step-by-Step Example
Question: In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find the length of the arc. (Use π = 22/7)
Solution:
- Identify the given values: Radius (r) = 21 cm Central angle (θ) = 60°
- Write the formula: Length of arc = (θ / 360) × 2πr
- Substitute the values: Length of arc = (60 / 360) × 2 × (22 / 7) × 21
- Simplify step by step:(60 / 360) simplifies to 1 / 6.Length of arc = (1 / 6) × 2 × (22 / 7) × 21Length of arc = (1 / 6) × 2 × 22 × 3Length of arc = (1 / 6) × 132Length of arc = 22 cm
Answer: The length of the arc is 22 cm.
📌 Area of a Sector
A complete circle covers 360° and has a total area of πr². A sector with central angle θ covers a fraction (θ / 360) of the complete circle area.
Sector Area Formula
Area of a sector = (θ / 360) × πr²
Where:
- θ = angle of the sector at the centre
- r = radius of the circle
- π = pi (22/7 or 3.14)
Plaintext
.--- Q ---.
/ \ Total Circle Area = πr²
/ O \
| /θ\ | Shaded Sector Area = (θ/360) × πr²
\ / r \ /
\ / \ / The sector area is a fraction of
A ---P--- B the total circular area.
Step-by-Step Example
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:
- Identify the given values: Radius (r) = 6 cm Angle (θ) = 60°
- Write the formula: Area of sector = (θ / 360) × πr²
- Substitute the values: Area of sector = (60 / 360) × (22 / 7) × 6 × 6
- Simplify step by step:(60 / 360) simplifies to 1 / 6.Area of sector = (1 / 6) × (22 / 7) × 36Area of sector = (22 / 7) × 6Area of sector = 132 / 7 square cm (or 18.86 square cm approx.)
Answer: The area of the sector is 132 / 7 cm².
🧠 Easy Way to Understand the Formula
Think of it like sharing a cake:
- The whole circle has 360° and total area = πr².
- If your slice has angle θ, your share of the cake is:Sector Area = (Required angle / 360) × Total Area of circle
If the angle is small (like 30°), the slice is small. If the angle is large (like 120°), the slice is large.
📌 Area of a Segment
To find the area of a minor segment, we look at the whole sector OAPB and remove the triangle OAB sitting inside it.
Segment Area Formula
Area of segment APB = Area of sector OAPB − Area of triangle OAB
Plaintext
[ SECTOR OAPB ] − [ TRIANGLE OAB ] = [ SEGMENT APB ]
.---. .---. .---.
/ \ / \ / \
| O | | O | | O |
\ /θ\ / \ / \ / \ /
'A---B' 'A---B' 'A===B'
We subtract the triangle because the sector contains both the triangle and the segment. Removing the triangle leaves only the curved segment region at the bottom.
Step-by-Step Example
Question: A chord of a circle of radius 10 cm subtends a right angle (90°) at the centre. Find the area of the corresponding minor segment. (Use π = 3.14)
Solution:
- Identify the given values: Radius (r) = 10 cm Central angle (θ) = 90°
- Step 1: Calculate the area of sector OAPB: Area of sector = (θ / 360) × πr² Area of sector = (90 / 360) × 3.14 × 10 × 10 Area of sector = (1 / 4) × 3.14 × 100 Area of sector = (1 / 4) × 314 = 78.5 cm²
- Step 2: Calculate the area of triangle OAB:Since central angle is 90°, triangle OAB is a right-angled triangle with base = 10 cm and height = 10 cm.Area of triangle = (1 / 2) × base × heightArea of triangle = (1 / 2) × 10 × 10 = 50 cm²
- Step 3: Subtract triangle area from sector area: Area of minor segment = Area of sector − Area of triangle Area of minor segment = 78.5 − 50 = 28.5 cm²
Answer: The area of the minor segment is 28.5 cm².
Major Segment
To find the area of a major segment, subtract the minor segment area from the total circle area:
Area of major segment = Area of complete circle − Area of minor segment
Area of major segment = πr² − Area of minor segment
Similarly, for a major sector: Area of major sector = Area of complete circle − Area of minor sector Or use formula: ((360 − θ) / 360) × πr²
🧠 How to Know Which Formula to Use?
Use this quick guide when solving exam questions:
- If the question asks for boundary length or arc length:
- Use Length of arc = (θ / 360) × 2πr.
- If the question asks for region enclosed by two radii (pizza slice):
- Use Area of sector = (θ / 360) × πr².
- If the question asks for region enclosed by a chord (cut region):
- Find Area of Sector = (θ / 360) × πr².
- Find Area of Triangle formed by radii and chord.
- Subtract: Segment Area = Sector Area − Triangle Area.
- If the question asks for a major region:
- Subtract the minor region from the total circle area (πr²).
🧠 Concepts Students Find Difficult
- Confusing sector and segment: Forgetting that sectors use radii while segments use chords.
- Confusing chord and radius: Mixing up line segments from the centre (radius) with lines connecting two boundary points (chord).
- Understanding minor and major regions: Not checking whether the question asks for the small region or the large remaining region.
- Choosing the correct angle: Using the given angle directly without checking if it refers to the minor or major part.
- Using 360 correctly: Making calculation errors while simplifying the fraction (θ / 360).
- Finding triangle area for a segment: Struggling to calculate triangle area when central angle is 60°, 90°, or 120°.
- Knowing when to add or subtract areas: Getting confused in combination problems (like table covers or shaded designs).
⚠️ Common Mistakes Students Make
- Using full circle area formula instead of sector formula:
- Mistake: Writing Area = πr² for a sector question.
- Fix: Always multiply πr² by (θ / 360).
- Forgetting to divide by 360:
- Mistake: Writing θ × πr² without dividing by 360.
- Fix: Double-check that (θ / 360) is written at the start of your formula.
- Confusing sector with segment:
- Mistake: Calculating sector area when the question specifically asks for segment area.
- Fix: Check if a chord is mentioned. If yes, remember to subtract the triangle area.
- Forgetting to subtract the triangle for segment questions:
- Mistake: Stopping after calculating the sector area.
- Fix: Remember: Segment = Sector − Triangle.
- Using diameter instead of radius:
- Mistake: Putting diameter value directly into formula r.
- Fix: Always check if given value is diameter. If diameter = 14 cm, then radius r = 7 cm.
- Writing wrong units:
- Mistake: Writing cm for area or cm² for arc length.
- Fix: Arc length is length (cm or m). Area is square units (cm² or m²).
- Using wrong central angle for major region:
- Mistake: Using θ = 60° when asked for major sector area.
- Fix: For major sector, use angle = (360° − 60°) = 300°, or subtract minor sector from total circle.
🔥 Priority Revision
⭐⭐⭐ HIGH-PRIORITY PRACTICE
- Area of minor and major sectors (Formula application).
- Area of minor segment when central angle is 60° or 90°.
- Clock hand questions (finding angle swept in given minutes).
⭐⭐ IMPORTANT PRACTICE
- Length of an arc of a sector.
- Finding segment area when central angle is 120°.
- Designs based on equal sectors (e.g., umbrella ribs, brooch wires, round table covers).
⭐ ADDITIONAL REVISION
- Finding grazing area of an animal tied to a peg.
- Area swept by car wipers or lighthouse beams.
- Multiple-choice questions on central angle formulas.
⚡ Areas Related to Circles Quick Revision Sheet
- Arc: Curved part of circle boundary.
- Sector: Region between two radii and an arc (Pizza slice).
- Segment: Region between a chord and an arc.
- Minor Region: Region with angle < 180°.
- Major Region: Region with angle > 180°.
- Arc Length Formula: Length = (θ / 360) × 2πr
- Sector Area Formula: Area = (θ / 360) × πr²
- Segment Area Method: Area = Sector Area − Triangle Area
- Major Sector Area: Total Circle Area − Minor Sector Area
- Major Segment Area: Total Circle Area − Minor Segment Area
- Units Reminder: Arc length in cm or m; Area in cm² or m².
📝 Test Yourself
- Identify whether the region enclosed by a chord and its corresponding arc is called a sector or a segment.
- A circle has radius 14 cm. An arc subtends an angle of 45° at the centre. Find the length of the arc.
- Find the area of a sector of a circle of radius 7 cm with a central angle of 90°.
- The minute hand of a wall clock is 21 cm long. Find the area swept by the minute hand in 10 minutes.
- A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor sector.
- For a circle of radius 10 cm with a central angle of 90°, calculate the area of the minor segment. (Use π = 3.14)
- Find the area of the major sector of a circle with radius 6 cm and minor sector angle of 60°. (Use π = 3.14)
- An umbrella has 6 ribs which are equally spaced. Assuming the umbrella to be a flat circle of radius 42 cm, find the area between two consecutive ribs.
- A wiper blade of length 18 cm sweeps through an angle of 120°. Calculate the total area cleaned by one sweep of the blade.
- A chord of radius 12 cm subtends 60° at the centre. Calculate the area of the minor segment. (Use π = 3.14 and √3 = 1.73)
🎯 Before Your Exam
☐ I understand what an arc is. ☐ I understand the difference between a sector and a segment. ☐ I know the difference between minor and major regions. ☐ I can find the length of an arc. ☐ I can find the area of a sector. ☐ I can find the area of a segment step by step. ☐ I know when to subtract triangle areas. ☐ I remember to write correct square units for area.
Stay focused, practise every calculation step by step, and you will achieve excellence in your board exam!