Areas Related to Circles is one of the most important chapters in Class 10 Maths because it connects circle geometry, formulas, sectors, segments, arcs, and shaded region questions in one place. Many students feel nervous about this chapter at first because it contains many related terms such as chord, tangent, secant, sector, segment, circumference, and arc. But once the basic structure is clear, the chapter becomes much easier.
This chapter is especially scoring when students understand the meaning of each part of a circle and know exactly which formula to apply in each question. Most mistakes happen not because the chapter is too difficult, but because students confuse sector with segment, arc with circumference, or area formulas with length formulas.
At Deeksha Vedantu, we always encourage students to learn this chapter in a concept-first manner. Once the definitions, formulas, and standard question types are understood properly, the chapter becomes very manageable and exam-friendly.
Chapter Overview at a Glance
This quick table helps students revise the full chapter faster.
Quick Concept Table
| Topic | Key idea |
| Circle | A closed figure whose boundary points are at equal distance from the center |
| Radius | Distance from center to any point on the circle |
| Diameter | Twice the radius |
| Circumference | Boundary length of the circle |
| Sector | Region formed by two radii and an arc |
| Segment | Region formed by a chord and an arc |
| Arc | Part of the circumference |
| Minor and major parts | Smaller and larger portions of sector, segment, or arc |
Introduction to Circle
Before solving questions from Areas Related to Circles, students should be clear about the meaning of a circle and the basic parts associated with it.
What Is a Circle
A circle is a closed figure in which every point on the boundary is at the same distance from one fixed point.
That fixed point is called the center of the circle, and the equal distance from the center to any point on the circle is called the radius.
This is the most basic idea of a circle and it helps students understand every other part of the chapter.
Important Basic Terms Related to Circle
Students should know these terms clearly before moving to sectors, segments, arcs, and shaded region questions.
Basic Circle Terms Table
| Term | Meaning |
| Center | Fixed point inside the circle |
| Radius | Line segment joining the center to any point on the circle |
| Diameter | Line segment passing through the center and joining two points on the circle |
| Circumference | Total outer boundary length of the circle |
| Chord | Line segment joining any two points on the circle |
| Tangent | Line touching the circle at exactly one point |
| Secant | Line cutting the circle at two points |
Relation Between Radius and Diameter
Diameter = 2 × radius
So,
d = 2r
Circumference of a Circle
The boundary or total outer length of a circle is called its circumference.
Formula for Circumference
Circumference = 2πr
Example
If radius = 7 cm, then:
Circumference = 2 × 22/7 × 7 = 44 cm
Area of a Circle
The area of a circle means the region enclosed inside the boundary of the circle.
Formula for Area of a Circle
Area = πr²
Example
If radius = 7 cm, then:
Area = 22/7 × 7 × 7 = 154 cm²
Why the Formula for Area of a Circle Works
A circle can be divided into many thin sectors. When these sectors are rearranged in alternate order, the figure starts looking like a rectangle.
Approximate Rectangle Idea
In that arrangement:
- length becomes πr
- breadth becomes r
So area becomes:
Area = πr × r = πr²
This gives the standard formula for the area of a circle.
Chord, Tangent, and Secant
These three terms are very important because students often confuse them in concept-based questions.
Chord of a Circle
A chord is a line segment joining any two points on the circle.
Important Note
Every diameter is a chord, but every chord is not a diameter.
This is because a diameter must pass through the center, while an ordinary chord does not need to.
Tangent to a Circle
A tangent is a line that touches the circle at exactly one point.
Important Idea
A tangent touches the circle, but does not cut through it.
From a Fixed External Point
From one fixed point outside the circle, exactly two tangents can be drawn.
Secant of a Circle
A secant is a line that intersects the circle at two points.
Difference Between Tangent and Secant
| Line type | How it meets the circle |
| Tangent | Touches the circle at one point |
| Secant | Cuts the circle at two points |
Common Tangents Between Two Circles
This is a useful board-level concept and is often asked in objective or concept-based form.
Case 1: Two Circles Are Separate
If the circles are completely separate, then 4 common tangents can be drawn.
Case 2: Two Circles Touch Each Other Externally
If the circles just touch each other, then 3 common tangents can be drawn.
Case 3: Two Circles Intersect Each Other
If the circles intersect, then 2 common tangents can be drawn.
Common Tangents Summary Table
| Position of circles | Number of common tangents |
| Separate circles | 4 |
| Touching externally | 3 |
| Intersecting circles | 2 |
Sector, Segment, and Arc of a Circle
These are the most important terms in this chapter. Students must understand them clearly because most questions are built around them.
Sector of a Circle
When two radii are drawn in a circle, the region between them is called a sector.
Types of Sector
| Type | Meaning |
| Minor sector | Smaller region formed by two radii |
| Major sector | Larger remaining region |
Easy Way to Remember
- smaller sector = minor sector
- larger sector = major sector
Segment of a Circle
When a chord is drawn in a circle, the circle is divided into two regions. These regions are called segments.
Types of Segment
| Type | Meaning |
| Minor segment | Smaller region between chord and corresponding arc |
| Major segment | Larger remaining region |
Arc of a Circle
An arc is a part of the circumference of a circle.
If two points are marked on the circle, then they divide the circumference into two arcs.
Types of Arc
| Type | Meaning |
| Minor arc | Smaller part of the circumference |
| Major arc | Larger part of the circumference |
Easy Difference Between Sector and Segment
| Figure part | Formed by |
| Sector | Two radii and an arc |
| Segment | A chord and an arc |
This difference is extremely important for the chapter.
Formula Sheet for Areas Related to Circles
This is the most important revision section of the chapter. Students should revise it regularly.
Main Formula Table
| Quantity | Formula |
| Circumference of circle | 2πr |
| Area of circle | πr² |
| Area of minor sector | (θ/360) × πr² |
| Area of major sector | ((360 – θ)/360) × πr² |
| Length of minor arc | (θ/360) × 2πr |
| Length of major arc | ((360 – θ)/360) × 2πr |
| Area of minor segment | Area of minor sector – area of triangle |
| Area of major segment | Area of circle – area of minor segment |
Formula Meaning Table
| Formula type | Used for |
| πr² | Area inside the full circle |
| 2πr | Boundary length of the full circle |
| (θ/360) × πr² | Area of a sector |
| (θ/360) × 2πr | Arc length |
Sector Formulas
These formulas are among the most used formulas in the chapter.
Area of Minor Sector
If θ is the central angle of the minor sector, then:
Area of minor sector = (θ/360) × πr²
Area of Major Sector
There are two ways to calculate it.
Method 1
Area of major sector = area of whole circle – area of minor sector
Method 2
Area of major sector = ((360 – θ)/360) × πr²
Both methods are correct.
Length of Minor Arc
If θ is the central angle, then:
Length of minor arc = (θ/360) × 2πr
This formula is used for boundary length, not area.
Length of Major Arc
Length of major arc = circumference of circle – length of minor arc
Or directly:
Length of major arc = ((360 – θ)/360) × 2πr
Segment Formulas
Segment-based questions are especially important for shaded region problems.
Area of Minor Segment
Area of minor segment = area of minor sector – area of triangle formed by the two radii and the chord
Area of Major Segment
Area of major segment = area of circle – area of minor segment
This is generally used when the larger region is asked.
Solved Question 1: Area of a Sector
Find the area of a sector of a circle of radius 21 cm and central angle 120°.
Given
- r = 21 cm
- θ = 120°
Step 1: Write the Formula
Area of sector = (θ/360) × πr²
Step 2: Substitute the Values
Area = (120/360) × 22/7 × 21 × 21
Step 3: Simplify
120/360 = 1/3
So,
Area = 1/3 × 22/7 × 21 × 21
= 462 cm²
Answer
Area of the sector = 462 cm²
Solved Question 2: Area of the Corresponding Major Sector
For the same circle of radius 21 cm and central angle 120°, find the area of the major sector.
Given
- r = 21 cm
- θ = 120°
Step 1: Use the Direct Formula
Area of major sector = ((360 – θ)/360) × πr²
Step 2: Substitute the Values
Area = ((360 – 120)/360) × 22/7 × 21 × 21
= (240/360) × 22/7 × 21 × 21
Step 3: Simplify
Area = 924 cm²
Answer
Area of the major sector = 924 cm²
Solved Question 3: Difference Between Areas of Minor and Major Sector
Find the difference between the areas of the two sectors when radius = 21 cm and central angle = 120°.
Given
- Minor sector area = 462 cm²
- Major sector area = 924 cm²
Step 1: Write the Difference
Difference = major sector area – minor sector area
Step 2: Substitute the Values
Difference = 924 – 462
Step 3: Simplify
Difference = 462 cm²
Answer
The difference between the two areas is 462 cm².
Solved Question 4: Length of Minor Arc
Find the length of the minor arc of a circle of radius 21 cm and central angle 120°.
Given
- r = 21 cm
- θ = 120°
Step 1: Write the Formula
Length of minor arc = (θ/360) × 2πr
Step 2: Substitute the Values
Length = (120/360) × 2 × 22/7 × 21
Step 3: Simplify
120/360 = 1/3
Length = 1/3 × 2 × 22/7 × 21
= 44 cm
Answer
Length of the minor arc = 44 cm
Solved Question 5: Length of Major Arc
Find the length of the major arc of the same circle.
Given
- r = 21 cm
- θ = 120°
- Minor arc length = 44 cm
Step 1: Find Total Circumference
Circumference = 2 × 22/7 × 21 = 132 cm
Step 2: Subtract Minor Arc Length
Length of major arc = 132 – 44 = 88 cm
Answer
Length of the major arc = 88 cm
How to Solve Shaded Region Questions
This is one of the most important exam areas from the chapter.
Step-by-Step Approach Table
| Step | What to do |
| Step 1 | Identify whether the shaded part is a sector, segment, arc, circle minus sector, sector minus triangle, or a combination |
| Step 2 | Write the correct formula before calculation |
| Step 3 | Break the figure into simple known shapes |
| Step 4 | Add or subtract areas carefully |
Common Shaded Region Patterns
Many shaded questions involve subtraction. For example:
- area of circle – area of sector
- area of sector – area of triangle
- area of major sector – area of minor sector
Board-Level Question Patterns from This Chapter
This chapter usually gives questions in a few repeated styles.
Board Pattern Summary Table
| Pattern | What students should focus on |
| Direct formula questions | Correct formula selection |
| Difference of areas | Careful subtraction |
| Shaded region questions | Figure breakdown |
| Segment-based questions | Sector area minus triangle area |
| Arc length questions | Use length formula, not area formula |
Common Mistakes Students Make in Areas Related to Circles
Common Mistakes Summary Table
| Mistake | Correct idea |
| Using πr² for arc questions | Arc length uses 2πr |
| Confusing sector and segment | Sector = two radii, segment = chord |
| Skipping 360 in formulas | Sector and arc formulas must be divided by 360 |
| Using wrong angle | Check minor and major parts carefully |
| Writing wrong units | Area in cm², length in cm |
Quick Revision Tips for Students
These tips are useful before exams.
Revision Strategy Table
| Step | What to do |
| Step 1 | Revise the full formula sheet daily |
| Step 2 | Draw the diagram before solving |
| Step 3 | Identify whether the question asks for area or arc length |
| Step 4 | Check whether the figure part is sector or segment |
| Step 5 | Practise shaded region questions separately |
Quick Self-Check Before Solving
Ask yourself:
- Is it area?
- Is it arc length?
- Is it sector or segment?
- Is it minor or major?
Practice Questions for Students
Important Practice Questions
- Find the circumference of a circle of radius 14 cm.
- Find the area of a circle of diameter 28 cm.
- Find the area of a sector of radius 14 cm and angle 90°.
- Find the length of the minor arc for a circle of radius 21 cm and angle 60°.
- Find the difference between the major and minor sector areas for a circle of radius 14 cm and angle 120°.
- Find the area of the minor segment when radius and central angle are given.
FAQs
Q1. What is the difference between a sector and a segment?
A sector is formed by two radii and an arc, while a segment is formed by a chord and an arc.
Q2. What is a minor arc?
The smaller part of the circumference between two points on a circle is called the minor arc.
Q3. What is the formula for the area of a sector?
The formula is (θ/360) × πr².
Q4. What is the formula for the length of an arc?
The formula is (θ/360) × 2πr for the minor arc.
Q5. How do I find the area of the major sector?
You can either subtract the minor sector from the total circle area or use the direct formula ((360 – θ)/360) × πr².
Q6. How do I find the area of a segment?
Find the area of the sector first, then subtract the area of the triangle formed inside it.
Q7. Why is Areas Related to Circles important for board exams?
It is important because it includes direct formula-based questions, shaded region problems, and application-based geometry questions.
Q8. How can I avoid mistakes in this chapter?
Always identify whether the question is asking for area or arc length, and whether the part is a sector or a segment.
Conclusion
Areas Related to Circles is a very important and scoring Class 10 Maths chapter when students understand the structure of the circle properly. Once the meanings of sector, segment, arc, tangent, secant, chord, and circumference become clear, the formulas start making much more sense. Most of the chapter then becomes a matter of choosing the correct formula and applying it carefully.
The best way to prepare this chapter is to revise formulas regularly, draw the figure before solving, and practise shaded region questions with patience. At Deeksha Vedantu, we always remind students that this chapter feels difficult only at the beginning. With concept clarity and repeated practice, it becomes one of the most comfortable chapters to score from.
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