त्रिज्यखंड और वृत्तखंड के क्षेत्रेफल

त्रिज्यखंड

चित्र 1 -त्रिज्यखंड

The circular region between two radii of a circle and the arc between them is called a sector of the circle. The sector always starts from the center of the circle. The semi-circle is also called the sector of the circle.

There are two types of sectors namely minor sector and major sector.

In Fig. 1 $OAPB$ is a sector of the circle with centre $O$. $\angle AOB$ is called the angle of sector. $OAPB$ is called the minor sector and $OAQB$ is called the major sector.

Angle of major sector is $360^{\circ }-\angle AOB$.

वृत्तखंड

चित्र 2 -वृत्तखंड

The part of the circular region enclosed between a chord and the corresponding arc is called a segment of the circle.

In Fig. 2 $AB$ is a chord of the circle with centre $O$.$APB$ is a segment of the circle.

There are two types of segment namely minor segment and major segment.

$APB$ is called the minor segment and $AQB$ is called the major segment.

त्रिज्यखंड का क्षेत्रफल

चित्र 3 -त्रिज्यखंड

Let us find the the area of a sector.

In the Fig 3 . Let $OAPB$ be a sector of a circle with centre $O$ and radius $r$ and $\angle AOB$ be $\theta$.

We know the area of a circle is $\Pi r^{2}$.

When the degree of measure of the angle at the centre is $360$, area of the sector = $\Pi r^{2}$

Hence when the degree of measure of the angle at the centre is $\theta$, area of the sector = ${\frac {\theta }{360}}\times \pi r^{2}$

Area of the sector of angle $\theta ={\frac {\theta }{360}}\times \pi r^{2}$ where is $r$ the radius of the circle and $\theta$ the angle of the sector in degrees.

Length and area of the arc $APB$ corresponding to the sector $OAPB$

चित्र 4 -त्रिज्यखंड

In the Fig 4.

When the degree of measure of the angle at the centre is $360$, length of the arc= $2\Pi r$

Hence when the degree of measure of the angle at the centre is $\theta$, length of the arc= ${\frac {\theta }{360}}\times 2\pi r$

Length of the arc = ${\frac {\theta }{360}}\times 2\pi r$

Area of the segment $APB$ = Area of the sector $OAPB$ - Area of the $\triangle OAB$

From Fig 3 and Fig 4

Area of the major sector $OAQB$ = $\Pi r^{2}$ – Area of the minor sector $OAPB$

Area of major segment $AQB$ = $\Pi r^{2}$ – Area of the minor segment $APB$

उदाहरण

उदाहरण- 1

In a circle of radius $21$ cm, an arc subtends an angle of $60^{\circ }$at the centre.

Find:

(i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord

Here $\theta =60$ $r=21$

(i) length of the arc $APB$ = ${\frac {\theta }{360}}\times 2\pi r$

=${\frac {60}{360}}\times 2\pi \times 21$ = $7\pi$ = $7\times {\frac {22}{7}}=22$ cm

(ii) area of the sector $OAPB$ = ${\frac {\theta }{360}}\times \pi r^{2}$

= ${\frac {60}{360}}\times \pi \times 21^{2}$ = $73.5\pi$ = $73.5\times {\frac {22}{7}}=231$ cm²

(iii) area of the segment $APB$ formed by the corresponding chord = area of the sector $OAPB$ - area of the triangle $OAB$

= $231-{\frac {\sqrt {3}}{4}}\times OB\times OA$

= $231-{\frac {\sqrt {3}}{4}}\times 21\times 21$

= $\left[231-{\frac {441{\sqrt {3}}}{4}}\right]$ cm²