What is the formula of area of minor sector?

Unless it is specifically requested, the $\pi$ value obtained by accessing the appropriate function in a scientific or graphic calculator should be used in any calculation involving ${\pi}$.

Table of Contents Show

Table of Contents
Circle and its Different Parameters
What is Area?
Area of a Circle
Area of Sector
Solved Examples – Area of a Sector
Frequently Asked Questions (FAQ) – Area of a Sector

ALL angles in this unit will be specified in degrees.

Measuring arc length

In any circle, the length of an arc ($l$) is proportional to the angle ($\theta$) it creates (subtends) at the centre.

For a circle of radius $r$, the circumference will be $2\pi$ units.

Thus,

$$\frac{\text{the length of the arc}}{\text{circumference}}=\frac{\theta}{360^{\circ}},$$

so $\begin{aligned}[t] &\frac{l}{2\pi r } = \frac{\theta}{360^{\circ}}.\\ \therefore l &= \frac{\theta}{360^{\circ}} \times {2\pi r}\\ &= \frac{\theta}{180^{\circ}} \times {\pi r}\\ &= {r} \times \frac{\pi}{180^{\circ}} \times {\theta} \text{ (Alternate formulations)} \end{aligned}$

A circle has a radius of 30 cm. Find the length of an arc subtending an angle of ${75^{\circ}}$ at the centre, correct to one decimal place.

Solution

$\begin{aligned}[t] \mbox{Length of arc}&=\frac{r\pi}{180^{\circ}}\times\theta\\ &=\frac{30\pi}{180^{\circ}}\times 75^{\circ}\\ &=39.2699\ldots\\ &\approx 39.3\text{ cm} \end{aligned}$

An arc of a circle has a length of 30 cm. If the radius of the circle is 25 cm, what angle, correct to the nearest whole degree, does the arc subtend at the centre?

Solution

$\begin{aligned}[t] 30&=\frac{r\pi}{180^{\circ}}\times {\theta}\\ &=\frac{25\pi}{180^{\circ}}\times {\theta}\\ \theta &=\frac{30 \times 180^{\circ}}{25 \times \pi}\\ &=68.754\ldots\approx {69^{\circ}} \end{aligned}$

Sector

The area of a sector depends not only on the radius of the circle involved, but also the angle between the two straight edges.

We can see this in the following table :

Angle	Fraction of circle area	Area rule
$180^{\circ}$	$\dfrac{180^{\circ}}{360^{\circ}}=\dfrac{1}{2}$	${A}=\dfrac{1}{2}\times {\pi r^2}$
$90^{\circ}$	$\dfrac{90^{\circ}}{360^{\circ}}=\dfrac{1}{4}$	${A}=\dfrac{1}{4}\times {\pi r^2}$
$45^{\circ}$	$\dfrac{45^{\circ}}{360^{\circ}}=\dfrac{1}{8}$	${A}=\dfrac{1}{8}\times {\pi r^2}$
$30^{\circ}$	$\dfrac{30^{\circ}}{360^{\circ}}=\dfrac{1}{12}$	${A}=\dfrac{1}{12}\times {\pi r^2}$
${\theta}$	$\dfrac{\theta}{360^{\circ}}$	${A}=\dfrac{\theta}{360}\times {\pi r^2}$

Find the area of a sector with a radius of 40 mm, that contains an angle of ${60^{\circ}}$, correct to the nearest square millimetre.

Solution

$$A=\frac{\theta}{360^{\circ}}\times {\pi r^2}=\frac{60^{\circ}}{360^{\circ}}\times {\pi 40^2}=923.628\ldots \approx 924\text{ mm}^2$$

Areas of segments

Where a chord subtends an angle ${\theta}$ at the centre of a circle of radius $r$, the area of the minor segment is given by:

\[\text{ Area} = \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right)\]

The area of the sector that includes the segment is given by $A=\dfrac{\theta}{360^{\circ}}\times {\pi r^2}$

or rewritten as

\[A = \dfrac{1}{2}\times {r^2} \times \dfrac{2 \pi \theta}{360^{\circ}}.\]

The area of the triangle whose boundaries are the two radii and the chord is given by

$\dfrac{1}{2}\times {r^2} \times {\sin \theta} \qquad \left(\text{from the Sine Area Rule} \quad \dfrac{1}{2}\times {ab} \sin {C}\right).$

Hence the area of the segment (minor) can be calculated by subtracting the area of the triangle from the area of the sector.

The area of the major segment can be calculated by taking the area of the minor segment from the total area of the circle.

Find the area, correct to two decimal places, of the minor segment in a circle of radius 10 cm where the angle subtended at the centre of the circle by the chord is ${85^{\circ}}$.

Solution

$\begin{aligned}[t] &\mbox{ Area}\\ &= \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right)\\ & = \frac{1}{2} \times {10^2} \left(\frac{2 \times \pi \times 85^{\circ}}{360^{\circ}} - {\sin {85}^{\circ}}\right) \\ &= 24.366\ldots \\ &\approx 24.37\text{cm}^2 \end{aligned}$

Find the area, correct to one decimal place, of the minor segment in a circle, of radius 15 cm where the chord length is also 15 cm.

Solution

If the chord length is the same as the radius of the circle, then the triangle that is formed will be equilateral (all three sides are 15 cm!), and the angle subtended at the centre by the chord will be ${60^{\circ}}$ (the angle inside an equilateral triangle).

$$ \text{Area} = \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right) = \frac{1}{2} \times {15^2} \left(\frac{2 \times \pi \times 60^{\circ}}{360^{\circ}} - {\sin 60^{\circ}}\right)= 20.381\ldots \approx 20.4\text{ cm}^2$$

Find the area, correct to two decimal places, of the minor segment in a circle, of radius 13 cm where the chord length is 21 cm.

Solution

To find the angle subtended at the centre by the chord, we will have to use the Cosine Rule

\begin{align*} A& = \cos^{-1} \left(\frac{b^2 + c^2 - a^2}{2bc}\right)\\ \mbox{ where} b& = c = \mbox{radius}\\ a& = \mbox{chord length}, \\ A&= \mbox{angle subtended at centre of circle}.\\ \text{So A}& = \cos^{-1} \left(\frac{13^2 + 13^2 - 21^2}{2 \times 13 \times 13}\right)\\ & = 107.74^{\circ} \end{align*} \begin{align*} \text{ Area}&= \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right)\\ &= \frac{1}{2} \times {13^2} \left(\frac{2 \times \pi \times 107.74^{\circ}}{360^{\circ}} - {\sin 107.74^{\circ}}\right)\\ &= 78.409\ldots\\ &\approx 78.41\text{ cm}^2 \end{align*}

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Written By Madhurima Das
Last Modified 26-10-2022

Area of a Sector: A sector is a pie-shaped portion of a circle. It can be compared to a slice of pizza. A sector is enclosed by two radii and an arc of a circle.

Every closed figure has some space within itself called its area. A sector of a circle is also a closed figure, and its area can be calculated using a unique formula. To discuss the area of a sector, we need to know what a circle is and its properties. An area of a sector involves few parameters of a circle such as a radius, arc, major sector, minor sector, etc.

Circle and its Different Parameters

A circle is the collection of all the points in a plane whose distance from a fixed point is always the same. The fixed point is called the centre of the circle, and the boundary of the circle is called the circumference of the circle.

Let us see the parts of a circle.

Radius

The radius$\left( r \right)$ of a circle is a line segment joining the centre and any point on the circumference.

Diameter

The diameter of a circle is a line segment starting from any point on the circumference of a circle, passing through the centre, and ending on the circumference at the opposite side of the circle. The length of the diameter is twice the length of the radius in a circle.

Arc

Any part of the circumference of a circle is called an arc of the circle. If the length of the arc of a circle is greater than a semicircle, it is called the major arc, and if the length of the arc of a circle is smaller than a semicircle, it is called a minor arc. The sum of lengths of the major arc and the minor arc will always give the circumference of the circle.

Sector

The area enclosed by an arc and the two radii joining the endpoints of the arc with the centre is called the sector of the circle.

When the sector is formed by the minor arc $PAQ,$ it is called the minor sector $POQA$ and when the sector is formed by the major arc $PBQ,$ it is called a major sector $PBQO.$

What is Area?

The area is the amount of space covered by a two-dimensional shape or surface. We measure the area in square units such as ${\rm{c}}{{\rm{m}}^2}$ or ${{\rm{m}}^2}.$

Area of a Circle

The area of a circle is defined by the space or region occupied by the circle in a two-dimensional plane.

Value of pi$\left( \pi \right)$:: A constant term “pi” is used in the formula of the area of a circle. $\pi $ is a constant term, also known as Archimedes constant. One way to define $\pi $ is that it is the ratio of the circumference of a circle to its diameter.

It is an irrational number whose value is $3.141592653589793238…$ For the common use in practice, the value of $\pi $ is approximately taken as $3.14$ when used as a decimal number and is taken as $\frac{22}7$ when used as a fraction to ease the calculation.

Formula for the Area of a Circle

If $r$ is the radius of a circle, then the area of a circle is given by $\pi {r^2}.$

Area of Sector

The region enclosed by a sector of a circle in a two-dimensional space is called the area of the sector, and the central angle $\theta$ between them is known as sector angle. This angle is known as the central angle.

The basic formula for the area of a circle, area $=\pi r^{2}$ can be applied to find the area of sectors of the circle

The full circle has an angle of $2 \pi$ radians around the centre. So, the area of the sector with a central angle $\theta$ and having radius $r$ will be proportional to this angle. The larger the angle $\theta ,$ the larger the area will be.

So, the area of the sector is given by,
Area $ = \frac{\theta }{{2\pi }} \times \pi {r^2} = \frac{\theta }{2} \times {r^2},$ where $\theta$ is in radians.

And, the area of the sector $ = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2} = \frac{{\theta \times \pi }}{{{{360}^{\rm{o}}}}} \times {r^2},$ where $\theta$ is in the degrees.

Units Used for Area of a Sector

The area of a sector is measured in square units. The unit used for the area of a sector in the CGS system is $\mathrm{cm}^{2}$ and in the SI system, the unit used is ${{\rm{m}}^2}.$

An Example of the Area of a Sector

The area of a sector can be explained by using one of the most common real-life examples of a slice of a pizza. The shape of slices of a circular pizza is like a sector. Each slice is a sector.

Solved Examples – Area of a Sector

Q.1. A circular arc whose radius is $6\,{\rm{cm}},$ makes an angle of ${45^{\rm{o}}}$ at the centre. Find the area of the sector formed. Use, $\pi = 3.14.$
Ans: Given that the radius of the circle is $6 \mathrm{~cm}$ and the angle between them is ${45^{\rm{o}}}.$The formula of the area of the sector $ = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2}$

So, the area of the sector formed $ = \frac{{{{45}^{\rm{o}}}}}{{{{360}^{\rm{o}}}}} \times 3.14{\left( 6 \right)^2} = 14.13\,{\rm{c}}{{\rm{m}}^2}$

Q.2. Find the area of the sector with a central angle of ${{{60}^{\rm{o}}}}$ and a radius of $9\,{\rm{cm}}$ using the value of $π=3.14.$
Ans: Given that the central angle is ${60^{\rm{o}}}$ and the length of the radius is $9\,{\rm{cm}}.$Central angle means the angle between two radii of a sector.The formula of the area of the sector $r = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2}$

So, the area of the sector formed $ = \frac{{{{60}^{\rm{o}}}}}{{{{360}^{\rm{o}}}}} \times 3.14{\left( 9 \right)^2} = 42.39\,{\rm{c}}{{\rm{m}}^2}$

Q.3. A circle-shaped badge is divided into $15$ sectors. The length of the diameter is $2\,{\rm{units}}{\rm{.}}$ Can you determine the area of each sector of the badge? Take the value of $π=3.14.$
Ans: Given, the diameter is $2\,{\rm{units}}.$The radius of the circular badge$ = \frac{2}{2} = 1\,{\rm{unit}}$The full circle has an angle ${360^{\rm{o}}}$ around the centre.Thus, $15$ sectors $ = {360^{\rm{o}}}$Hence, the central angle of each sector $ = \frac{{{{360}^{\rm{o}}}}}{{15}} = {24^{\rm{o}}}$The formula of the area of the sector$ = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2}$

So,the area of the sector $ = \frac{{{{24}^{\rm{o}}}}}{{{{360}^{\rm{o}}}}} \times 3.14 \times {\left( 1 \right)^2} = 0.209\,{\rm{uni}}{{\rm{t}}^2}.$

Q.4. $PQ$ is a chord of a circle that subtends an angle of ${{{60}^{\rm{o}}}}$ at the centre of a circle. The radius of the circle is $6$ inches. Can you find the area of the minor sector of this circle? Use $π=3.14.$
Ans: The radius of the circle is $6$ inches. We will use the formula of the area of a sector of a circle.The area of minor sector $ = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2} = \frac{{{{60}^{\rm{o}}}}}{{{{360}^{\rm{o}}}}}3.14 \times {\left( 6 \right)^2} = 0.209\,{\rm{uni}}{{\rm{t}}^2}$

Therefore, the area of the minor sector is $18.84\,{\rm{uni}}{{\rm{t}}^2}.$

Q.5. An umbrella has equally spaced $8$ ribs. If viewed as a flat circle of radius $14\,{\rm{units}},$ then what would be the area between two consecutive ribs of the umbrella. Use $\pi = \frac{{22}}{7}.$
Ans: The radius of the flat umbrella would be $14\,{\rm{units}}{\rm{.}}$There are $8$ ribs in the umbrella.The angle of each sector of the umbrella is $45°$ as360°45°Thus, the area of sector $ = \frac{\theta }{{{{360}^{\rm{o}}}}} \times \pi {r^2} = \frac{{{{45}^{\rm{o}}}}}{{{{360}^{\rm{o}}}}} \times \frac{{22}}{7} \times {\left( {14} \right)^2} = 77\,{\rm{uni}}{{\rm{t}}^2}$

Therefore, the area between the two consecutive ribs of the umbrella is $77\,{\rm{uni}}{{\rm{t}}^2}.$

Summary

A sector is a closed shape enclosed by the radii of a circle and an arc of it. The area of a sector is the space occupied by the sector. We have learned in this article how to find the area of a sector when the central angle is given. We also discussed some important parts of the circle such as radius, minor arc, major arc, minor sector, major sector, etc. So, we can say that an area of a sector is the fraction or a part of the area of a circle.

Learn Everything About Circles Here

Frequently Asked Questions (FAQ) – Area of a Sector

Q.1. What is the area of the major sector?
Ans: If the central angle of a sector(minor sector) is $θ$ then, the formula of the major sector is $=\frac{360^{\circ}-\theta}{360^{\circ}} \times \pi r^{2}$ where r is the radius of the circle.
Area of major sector $=$ Area of Circle $ – $ Area of minor sector

Q.2. What is the formula for the area of a sector?
Ans: The formula of the area of sector $=\frac{\theta}{360^{\circ}} \times \pi r^{2}$ where $r$ is the radius of the circle.

Q.3. What is a perimeter of a sector?
Ans: A sector is enclosed by two line segments (radii of the circle) and an arc of a circle. An arc is a part of the circumference of a circle.
The perimeter of the sector$ = 2 \times {\rm{radius}} + {\rm{Arc}}\,{\rm{length}} = 2r + \frac{\theta }{{{{360}^{\rm{o}}}}} \times 2\pi r.$

Q.4. What is a sector in circles?
Ans: The area enclosed by an arc and the two radii joining the endpoints of the arc with the centre is called the sector of the circle.

Q.5. What is the area of the minor sector?
Ans: If the central angle of the minor sector is $θ$ then, the formula of the minor sector is $=\frac{\theta}{360^{\circ}} \times \pi r^{2}$ where $r$ is the radius of the circle.

Now that you are provided with all the necessary information about the area of a sector and its formulas, we hope this article is helpful to you. If you have any queries on this page, post your comments in the comment box below and we will get back to you as soon as possible.

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