# Complete Guide For Class 10 Math Chapter 11 – Areas Related to Circles

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The chapter Areas Related to Circles explores the concepts related to the measurement of areas in circular figures. Students learn to calculate the area of a circle using the formula πr², where r is the radius. The chapter also covers the concepts of the circumference of a circle, sectors, and segments. Key formulas like the area of a sector (θ/360 × πr²) and the area of a segment are introduced. These concepts are crucial for solving problems related to circular regions, including real-life applications like finding the area of land, designing circular objects, and more. Mastery of these topics enhances students’ ability to handle complex geometrical problems involving circles and their parts.

## Areas Related to Circles

A circle is a basic shape in Euclidean geometry consisting of points on a plane that are equidistant from a fixed point (the center). The distance from the center to any point on the circle is called the radius.

### Radius and Circumference of a Circle

The circumference of a circle is the distance covered by traveling once around the circle. It is the perimeter of the circle, and this distance bears a constant ratio with the diameter of the circle. This constant ratio is denoted by π (pi), which approximately equals 22/7 or 3.14.

The formula for the circumference of a circle is given by: **Circumference = 2πr**, where r is the radius of the circle.

### Area of a Circle

To understand the area of a circle, imagine cutting a circular object (like a sheet) into several sectors and rearranging them. This arrangement forms a shape that resembles a rectangle. The area of the circle is equal to the area of this rectangle.

The formula for the area of a circle is **Area = πr²**, where r is the radius of the circle.

### Sector and Segment of a Circle

- A sector of a circle is the portion enclosed by two radii and the corresponding arc.
- A segment of a circle is the portion enclosed between a chord and the corresponding arc.

## Related Terms For Areas Related To Circles

**Minor Sector**: The smaller part of the circle when it is divided by a chord.**Major Sector**: The larger part of the circle when it is divided by a chord.**Minor Segment**: The area enclosed by the chord and the minor arc.**Major Segment**: The area enclosed by the chord and the major arc.

When we refer to a “segment” or “sector,” we usually mean the “minor segment” or “minor sector.”

**Area of a Sector**

The area of a sector can be calculated using the formula: Area of Sector = θ / **360 × πr**², where θ is the angle subtended by the sector at the center of the circle.

When the degree measure of the angle at the center is 360,

area of the sector = πr²

So, when the degree measure of the angle at the center is 1,

area of the sector = πr²/360

Therefore, when the degree measure of the angle is θ

area of the sector = πr²/360 x θ

= θ / 360 x πr²

Thus, the area of the sector of angle θ = θ / 360 x πr²

**Length of an Arc**

The length of an arc of a sector with angle θ can be calculated using: **Length of Arc = θ /360 × 2πr**

**Example**:

Find the length of the arc of a circle of radius 21 cm, if the angle subtended by the arc at the center is 60°.

Length of the arc = 2πr x θ/ 360 = 2 x 22 x 21 x 60 / 7 x 360 = 22 cm

**Area of Segments of a Circle**

The area of a segment can be calculated as** Area of Segment = Area of Sector − Area of Triangle**

Area of segment APB = Area of the sector OAPB – Area of ∆OAB

= πr²θ/ 360 – Area of ∆OAB

Area of major sector OAQB = πr² – Area of minor sector OAPB

Area of major sector AQB = πr² – Area of minor sector APB

**Example**: Given a chord of a circle with a radius of 15 cm that subtends an angle of 60° at the center, find areas of corresponding minor sectors and segments of a circle.

Area of the sector = πr²θ/ 360

Area of the segment = πr²θ/ 360 – Area of ∆OAB

In ∆OAB, ∠AOB = 60⁰ (given)

OA = OB (Raddi of the same circle)

⇒ ∠OAB = ∠OBA = 180⁰- 60⁰ /2 = 60⁰ each

⇒ ∆OAB is an equiangular.

⇒ ∆OAB is an equilateral triangle.

We have, θ = 60⁰

Hence, are of the sector = π(15²) 60 / 360 = π(15²)/6 = 117.86 cm²

Given ∆OAB is an equilateral triangle whose side is equal to the radius of the circle.

So, area of the segment = π(15²) 60 / 360 – √3 /4 (15²)

= (15²) (π/6 – √3 /4) = 20.42 cm²

## Let’s Conclude

In conclusion, CBSE Class 10th Math, Chapter 11 – *Areas Related to Circles*, provides a thorough understanding of key concepts related to circular figures. From calculating the area of a circle using πr² to understanding sectors, segments, and arcs, this chapter equips students with essential geometry skills. Mastering these concepts is not only crucial for academic success but also for solving real-world problems involving circular shapes. With iPrep’s comprehensive learning resources, including animated videos, practice questions, and detailed notes, students can confidently excel in *Areas Related to Circles*. Explore our engaging content to enhance your learning and solidify your understanding of *Areas Related to Circles*.

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