Introduction to Areas Related to Circles

The chapter Areas Related to Circles is an important part of Class 10 Mathematics. In earlier classes, students learned basic concepts about circles such as radius, diameter, circumference, and area. In this chapter, students learn how to calculate the area and perimeter of different parts of a circle, especially sectors and segments.

This chapter is practical and formula-based, which means if students understand the formulas and practice numerical questions regularly, they can score good marks easily.

Circles are used in many everyday objects such as:

• Wheels
• Clocks
• Coins
• Rings
• Plates
• Circular Parks

Understanding areas related to circles helps in solving real-life measurement problems.

This chapter mainly covers:

• Circumference of a Circle
• Area of a Circle
• Sector of a Circle
• Arc Length
• Area of Sector
• Segment of a Circle
• Area of Segment

Most board exam questions in this chapter are numerical and formula-based.

What is a Circle?

A circle is a closed figure in which every point on the boundary is at an equal distance from a fixed point called the centre.

The fixed distance from the centre to the boundary is called the radius.

Important Terms of Circle

Before understanding formulas, students should know some basic terms.

Centre

The fixed point inside the circle.

Example: O

Radius

The line joining the centre to any point on the circle.

Represented as: r

Diameter

A line passing through the centre and joining two points of the circle.

Relationship:

Circumference

The boundary or outer edge of the circle.

Chord

A line joining two points on the circle.

Arc

A portion of the circumference of a circle.

Sector

A region enclosed by two radii and one arc.

Segment

The area between a chord and its corresponding arc.

These concepts are very important for solving problems.

Circumference of a Circle

The circumference is the total distance around a circle.

Formula:

Where:

C = Circumference
r = Radius
π = 22/7 or 3.14

Example

If:

Radius = 7 cm

Then:

Circumference = 2 × 22/7 × 7 = 44 cm

Therefore, the circumference is 44 cm.

Area of a Circle

The area of a circle means the total region enclosed inside the boundary.

Formula:

Where:

A = Area
r = Radius

Example

Radius = 7 cm

Area = 22/7 × 7 × 7 = 154 cm²

Thus, the area of the circle becomes 154 cm².

Students should remember this formula carefully.

What is a Sector of a Circle?

A sector is a portion of a circle enclosed between two radii and one arc.

There are two types of sectors.

Minor Sector

The smaller region of the circle.

Major Sector

The larger region of the circle.

Sectors are mainly identified using the angle formed at the centre.

Length of an Arc

The curved part of a circle is called an arc.

Formula for arc length:

Where:

θ = Angle at the centre

Example

Suppose:

Radius = 14 cm
Angle = 90°

Arc Length = (90/360) × 2 × 22/7 × 14 = 22 cm

Therefore, the arc length becomes 22 cm.

Area of Sector

The area of a sector depends on the angle at the centre.

Formula:

Where:

θ = Central Angle
r = Radius

Example

Suppose:

Radius = 7 cm
Angle = 90°

Area = (90/360) × 22/7 × 7² = 38.5 cm²

Hence, the area of the sector is 38.5 cm².

This formula is frequently used in examinations.

What is a Segment of a Circle?

A segment is the region between a chord and its corresponding arc.

There are two types.

Minor Segment

The smaller part of the circle.

Major Segment

The larger part of the circle.

Segment questions are slightly difficult because students must subtract areas.

Area of Segment

Formula:

Area of Segment = Area of Sector − Area of Triangle

This means:

Example

Suppose:

Sector Area = 80 cm²
Triangle Area = 30 cm²

Then:

Segment Area = 80 − 30 = 50 cm²

Thus, the segment area becomes 50 cm².

Students should practice this concept carefully.

Combination Problems

In board exams, questions often involve combined shapes.

Examples:

• Semi-circle + Rectangle
• Circle − Square
• Sector + Triangle

Students should solve these by dividing the figure into smaller parts.

Semi-Circle Formula

Area of Semicircle:

Circumference of Semicircle:

Quarter Circle Formula

Area:

These formulas are important.

Applications of Areas Related to Circles

Construction

Engineers calculate circular land areas.

Architecture

Circular buildings and parks require area calculations.

Sports

Circular grounds use circle formulas.

Manufacturing

Machines often contain circular parts.

Transportation

Wheel dimensions involve circumference calculations.

This proves circles are useful in practical life.

Common Mistakes Students Make

1. Wrong Formula Usage

Students often confuse area and circumference formulas.

Remember:

Circumference → Around the Circle
Area → Inside the Circle

2. Forgetting Units

Always write:

• cm² for Area
• cm for Length

3. Calculation Mistakes

Simplify π carefully.

4. Incorrect Angle in Sector Formula

Use the correct central angle.

5. Segment Formula Errors

Remember:

Segment = Sector − Triangle

Not addition.

Tips to Score Good Marks

• Learn formulas properly.

• Practice numerical questions daily.

• Understand sector and segment clearly.

• Solve NCERT examples multiple times.

• Revise π calculations regularly.

This chapter is highly scoring with practice.

Conclusion

The chapter Areas Related to Circles helps students understand how to calculate the area and perimeter of different parts of a circle. Concepts such as circumference, area of a circle, sector, arc length, and segment area are extremely important for board examinations.

Although the formulas may initially seem confusing, regular practice makes the chapter easy and scoring. Students should focus on understanding formulas instead of memorizing them blindly. With sufficient practice, this chapter becomes one of the highest-scoring topics in Class 10 Mathematics.