AREAS RELATED TO CIRCLES-Notes

Sector

sector

A major sector of a circle is the larger region formed when two radii of a circle divide it into two unequal parts. When the angle between the two radii at the centre is greater than \(180^\circ\), the sector corresponding to this angle is called the major sector. The remaining smaller part, with an angle less than \(180^\circ\), is known as the minor sector.

The concept of a major sector is closely related to that of a minor sector, as both together make up the entire circle. Since the total angle around the centre of a circle is \(360^\circ\), the angle of the major sector can be found by subtracting the angle of the minor sector from \(360^\circ\). Because the area of a sector is directly proportional to its central angle, the major sector always occupies more than half of the area of the circle.

To find the area of a major sector, we usually calculate the area of the whole circle and then subtract the area of the corresponding minor sector. This approach simplifies calculations and avoids dealing with large central angles directly. The formula reflects the idea that the major sector is simply what remains after removing the smaller sector from the circle.

\[\tiny\begin{aligned} \color{blue} \boxed{\boldsymbol{\textbf{Area of the sector of angle }\theta \\= \frac{\theta}{360}\,\times\pi r^{2}}} \end{aligned}\]
\[\tiny\begin{aligned} \color{blue} \boxed{\boldsymbol{\textbf{Length of an arc of a sector of angle }\theta \\= \frac{\theta}{360}\,\times 2\pi r}} \end{aligned}\]

Segment

segment

A segment of a circle is the region formed when a chord divides the circle into two parts. The chord is a straight line joining any two points on the circumference of the circle, and the curved boundary of the segment is the arc corresponding to that chord. Unlike a sector, a segment does not involve radii directly, but it is closely connected to both circular and triangular geometry.

There are two types of segments: the minor segment and the major segment. The smaller region bounded by the chord and the shorter arc is called the minor segment, while the larger region enclosed by the chord and the longer arc is known as the major segment.

To find the area of a segment, we first consider the sector formed by joining the endpoints of the chord to the centre of the circle. The area of the triangle formed by the two radii and the chord is then subtracted from the area of this sector. This difference gives the area of the minor segment. For the major segment, the area is obtained by subtracting the area of the minor segment from the area of the entire circle.

Example-1

Fig. 11.5-x
Fig. 11.5

Find the area of the sector of a circle with radius 4 cm and an angle of \(\angle 30°\). Also, find the area of the corresponding major sector (Use \(\pi\) = 3.14).

Solution:

Radius = 4cm
Angle of minor sector = 30°
Hence, Angle of major sector = 360°-30° = 330°

Area of major sector (A)

$$\begin{aligned} A&=\dfrac{\theta }{360^{\circ }}\cdot \pi r^{2}\\\\ &=\dfrac{330^\circ}{360^\circ}\times 3.14\times 4\times 4\\\\ &\approx 46.1\ cm^{2}\end{aligned}$$
Area of the major sector is \(\approx 46.1\ cm^{2}\)

Example-2

Fig. 11.6-x
Fig. 11.6

Find the area of the segment AYB shown in Fig. 11.6, if the radius of the circle is 21 cm and \(\angle AOB = 120°.\) (Use \(\pi = \frac{22}{7} \))

Solution:

Angle of minor sector = 120°
Radius of the circle = 21cm
Area of the sector \((A_s)\)

$$\require{cancel}\begin{aligned}A_s&=\dfrac{\theta }{360}\times \pi r^{2}\\ &=\dfrac{120}{360}\times \dfrac{22}{7}\times 21^2 \\ &=\dfrac{1}{\cancelto{1}3}\times \dfrac{22}{\cancelto{1}7}\times \cancelto{\cancelto{1}3}{21}\times 21\\ &=22\times 21\\ &=462\ cm^{2}\end{aligned}$$

Consider triangle OAB
OA B is an isosceles triangle, angle OAD

Fig. 11.6.1-x
Fig. 11.6.1

$$\small\begin{aligned}\angle OAB+\angle OBA&+\angle AOB=180^{\circ }\\ \angle OAB&=2\angle OBA\\ \therefore 2\angle OAB&=180-\angle AOB\\\\ &=\dfrac{180^{\circ }-120}{2}\\ &=60\\ \Rightarrow \angle OAB&=30^{\circ }\\\\ \sin 30^\circ &=\dfrac{OD}{OA}\\ OD&=OA\cdot \sin 30^{^\circ}\\ &=21\times \dfrac{1}{2}\\ &=\dfrac{21}{2}\\\\ \cos 30^{^\circ}&=\dfrac{AD}{OA}\\ AD&=OA\cdot \cos 30^{\circ }\\ AD&=\dfrac{\sqrt{3}}{2}-21\\ AB&=2AD\\ &=\dfrac{\sqrt{3}}{2}\cdot 21\cdot 2\\ &=21\sqrt{3}\\\\ A_{t}&=\dfrac{1}{2}AB\cdot OD\\\\ &=\dfrac{1}{2}\cdot 21\sqrt{3}\cdot \dfrac{21}{2}\\\\ &=\dfrac{441\sqrt{3}}{2\cdot2}\\\\ &=\dfrac{441}{4}\sqrt{3}\end{aligned}$$

Area of segment \(A_{sg}\) = Area of sector-Area of triangle OAB

$$\begin{aligned}A_{sg}&=462-\dfrac{441\sqrt{3}}{4}\\\\ &=\dfrac{1848-441\sqrt{3}}{4}\\\\ &=\dfrac{21}{4}\left( 88-21\sqrt{3}\right) \end{aligned}$$

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    Frequently Asked Questions

    A circle is the locus of all points in a plane that are at a fixed distance, called the radius, from a fixed point known as the centre.

    The area of a circle is the region enclosed by its circumference and is calculated using the formula \(A = \pi r^2\).

    \(\pi\) is a constant representing the ratio of the circumference of a circle to its diameter, commonly taken as \(\frac{22}{7}\) or 3.14.

    A sector is the region bounded by two radii and the arc between them.

    A minor sector is the smaller sector formed when the central angle is less than \(180^\circ\).

    A major sector is the larger sector formed when the central angle is greater than \(180^\circ\).

    The area of a sector is \(\frac{\theta}{360^\circ} \times \pi r^2\), where \(\theta\) is the central angle.

    The angle at the centre determines what fraction of the circle the sector occupies, directly affecting its area.

    A segment is the region bounded by a chord of a circle and the corresponding arc.

    A minor segment is the smaller region formed between a chord and the corresponding minor arc.

    A major segment is the larger region formed between a chord and the corresponding major arc.

    Area of minor segment = Area of corresponding sector - Area of the triangle formed by the radii and chord.

    Area of major segment = Area of the circle - Area of the minor segment.

    A triangle helps remove the straight-line portion inside the sector, leaving only the curved region of the segment.

    A chord is a line segment joining any two points on the circumference of a circle.

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