Area of the segment and sector of a circle - Future Study Point

# Segment and sector of a circle

You are familiar with some methods of finding the area of plane figures like triangle, square, rectangle, parallelogram, rhombus, and of the circle. Here the area of segment and sector of the circle is of great importance in our day to life like areas of the rounded figure as an example, wheel excel, and other rounded tools and machines. The segment and sector are the part of the rounded figure are needed in manufacturing different types of tools used in machines and architectural designs of bridges and buildings, so the area of segments and sectors of the circle is introduced in the 10-grade syllabus of every school board.

NCERT Solutions Class 10 Science from chapter 1 to 16

Area of the segment and sectors is the topic referred from chapter 12 of the class 10 NCERTย  book prescribed by CBSE a reputed board of India for school education.

## Sector of circle :

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The area covered by two radii and the corresponding arc of the circle is known as sector.

360ยฐ angle around the centre corresponds to the areaย $= \pi r^{2}$

1ยฐ angle around the circle corresponds to the areaย  $= \frac{\pi r^{2}}{360}$

ฮธ angle around the circle correspond to the areaย  $= \frac{\pi r^{2}}{360}\times \Theta$

Hence the area of a sector

$= \frac{\Theta }{360}\times \pi r^{2}$

Where ฮธ is the angle between two radii ofย  the circle.

Type of sectors : Sectors of circles are of two kinds (a) Minor sector (b) Major sector

Minor sector: The shaded regionย  in above figure is minor sector i.e The smaller sector is known as minor sector.

Major sector: The unshaded regionย  in above figure is major sector i.e The larger sector is known as major sector.

Area of major sector = Area of circle – area of minor sector

$Area \: of\: major \: sector =\pi r^{2}-\frac{\Theta }{360}\pi r^{2}$

## Segment of a circle :

The area covered by the chord of a circle and corresponding arc known as segment.

Area of segment = area of sector – area of triangle covered by two radii and corresponding chord of the circle

Let the angle subtended by the two radii is = ฮธ

โ AOM = ฮธ/2 (OMโฅ AB)

$sin \frac{\Theta }{2} =\frac{AM}{r}$

AM = r sinฮธ/2

AB = 2AM=2r sinฮธ

$cos \frac{\Theta }{2} =\frac{OM}{r}$

OM = r cosฮธ/2

Area of ฮAOB

$=\frac{1}{2}AB\times OM$

$=\frac{1}{2}2rsin\frac{\Theta }{2}.rcos\frac{\Theta }{2}$

Since ,2sin ฮธ cos ฮธ =sin2ฮธ

$=\frac{1}{2}r^{2}sin\Theta$

Hence area of ฮAOB = 1/2 rยฒ sinฮธ

We know area of sector

$= \frac{\Theta }{360}\times \pi r^{2}$

So, area of segment

$=\frac{\Theta }{360}\pi r^{2}-\frac{1}{2}r^{2}sin\Theta$

$=\frac{r^{2}}{2}\left ( \frac{\pi \Theta }{180}-sin\Theta \right )$

Hence Area of segment

$=\frac{r^{2}}{2}\left ( \frac{\pi \Theta }{180}-sin\Theta \right )$

## Type of segment

(a) Minor segment: The smaller segment of the circle is known as minor segment i.e The shaded region in above figure is known as minor segment.

(b) Major segment: The larger segment of the circle is known as major segment i.e The unshaded region in above figure is known as major segment.

The area of larger segment = Area of circle – area of minor sector

$Area \: of \: major \: segment=\pi r^{2}-\frac{r^{2}}{2}\left ( \frac{\pi \Theta }{180}-sin\Theta \right )$

## Length of arc :

Let the length of arc corresponding to two radii of circle is l.

360ยฐ angle corresponds to $=2\pi r$

1ยฐ angle corresponds toย  ย  $=\frac{2\pi r}{360}$

ฮธ angle corresponds toย  ย $=\frac{2\pi r}{360}\times \Theta$

Hence the length of arc corresponding to two radiiย  ย $l= \frac{\Theta }{360} \times 2\pi r$

## Type of Arc

(a) Minor arc: The smaller arc of the circle is known as minor arc, in the figure l is the length of minor arc.

$l=\frac{\Theta }{360^{0}}\times 2\pi r$

(b) Major arc: The larger arc of the circle is known as major arc

The length of major arc = Circumference of the circle – The length of minor arc

$The \: length \: of \: major\: arc=2\pi r-\frac{\Theta }{360}\times 2\pi r$

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