NOTES

📘 Areas Related to Circles Class 10 Notes (Chapter 11)

🔷 Introduction

Areas related to circles is an important chapter in geometry. In simple words, this chapter helps us find the area and perimeter of circular shapes. Therefore, students learn how to solve many practical problems.

Moreover, circles appear in wheels, coins, plates, gardens, and tracks. As a result, this chapter connects mathematics with daily life.

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🔷 Important Terms

Before solving questions, understand these terms carefully. Thus, formulas become easier to remember.

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✔ Circle

A circle is a round figure where every point stays at the same distance from the center.

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✔ Radius

A line from the center to any point on the circle is called the radius.

Represented by:$$r$$r

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✔ Diameter

A line passing through the center and joining two points of the circle is called the diameter.

Formula:$$d=2r$$d=2r

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✔ Circumference

The boundary of a circle is called circumference.

Formula:$$C=2\pi r$$C=2πr

or$$C=\pi d$$C=πd

Therefore, circumference gives the total boundary length.

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🔷 Area of a Circle

The area of a circle shows the space inside it.

Formula:$$A=\pi r^2$$A=πr2

Where:

• $\pi=\frac{22}{7}$π=722​ or 3.14
• $r$r= radius

Thus, this formula appears in almost every question.

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Example

Find the area of a circle with radius 7 cm.

Solution

Given:$$r=7cm$$r=7cm

Now use:$$A=\pi r^2$$A=πr2 $$A=\frac{22}{7}\times7\times7$$A=722​×7×7 $$A=154cm^2$$A=154cm2

Answer:

Area = 154 cm²

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🔷 Arc of a Circle

A part of the boundary of a circle is called an arc.

Moreover, an arc may be small or large.

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🔷 Sector of a Circle

A sector forms when two radii and one arc enclose a region.

For example, a pizza slice looks like a sector.

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🔷 Area of a Sector

Formula:$$Area=\frac{\theta}{360^\circ}\times\pi r^2$$Area=360∘θ​×πr2

Where:

• $\theta$θ= central angle
• $r$r= radius

Therefore, larger angles create larger sectors.

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Example

Find the area of a sector of angle 90° and radius 14 cm.

Solution

Given:$$\theta=90^\circ$$θ=90∘ $$r=14cm$$r=14cm

Now use:$$Area=\frac{90}{360}\times\pi\times14^2$$Area=36090​×π×142 $$=\frac14\times\frac{22}{7}\times196$$=41​×722​×196 $$=154cm^2$$=154cm2

Answer:

Area = 154 cm²

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🔷 Length of an Arc

Formula:$$Length=\frac{\theta}{360^\circ}\times2\pi r$$Length=360∘θ​×2πr

Thus, arc length gives the curved boundary.

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🔷 Segment of a Circle

A chord and an arc together form a segment.

In addition, a segment may be minor or major.

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🔷 Area of a Segment

Formula:$$Area\ of\ Segment= Area\ of\ Sector-Area\ of\ Triangle$$Area of Segment=Area of Sector−Area of Triangle

Therefore, solve both parts separately.

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🔷 Area of a Semicircle

A semicircle is half of a circle.

Formula:$$Area=\frac12\pi r^2$$Area=21​πr2

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🔷 Perimeter of a Semicircle

Formula:$$Perimeter=\pi r+2r$$Perimeter=πr+2r

Hence, include the diameter while finding perimeter.

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🔷 Area of a Quadrant

A quadrant is one-fourth of a circle.

Formula:$$Area=\frac14\pi r^2$$Area=41​πr2

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🔷 Important Properties

Remember these points carefully.

• Diameter = 2 × Radius
• Circumference = 2πr
• Area = πr²
• Sector area depends on angle
• Arc length depends on angle

Therefore, these formulas help in most exam questions.

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🔷 Real-Life Uses

People use these formulas in many fields. For example, they help in:

• Designing parks
• Building roads
• Making wheels
• Sports tracks
• Architecture
• Engineering

Thus, circles have many practical uses.