Area of Sector of a Circle

Introduction

The area of sector of a circle is the amount of area enclosed within the sector boundary. In this tutorial, we will learn about the sectors of a circle, and the area of the sector of a circle.

The sector area of a circle is the amount of area enclosed within the sector boundary.

A sector always begins from the centre of the circle. An area of a sector is defined as the element of the circle bounded between two radii and their adjoining arcs. A semicircle is the most frequent sector of a circle that represents a semicircle. Learn more about area, its formula, and how to calculate sector area by the use of radians and degrees in the next portions of this article.

We will also solve some examples to gain a better understanding of these topics.

Sector of a Circle

A sector is considered a portion of a circle with two radii and one arc. The circle is divided into two sectors, a minor sector and a major sector. The minor area is the smaller part of the circle, and the larger section is the major part.

Area of a sector of a circle

• Use the sector formulation area to locate the whole area enclosed by using the sector. The area of the sector can be calculated the use of the formula −

  $$\mathrm{Circular area =\frac{θ}{360^\circ} × πr^2}$$

  where θ is the angle (in angles) at the centre and 'r' the radius of the circle.

• Circle Sector Area =$\mathrm{\frac{1}{2} × r^2 θ }$, where θ is the angle subtended through the central arc in radians and 'r' is the radius of the circle.

• Let's derive the components for the area of a sector of a circle. We know that an ideal circle is 360 degrees. The area of a circle with an angle of 360° at its centre is given via πr², where 'r' is the radius of the circle.

• If the angle at the centre of the circle is 1°, the area of the area is $\mathrm{\frac{ πr^2}{360°}}$. Therefore, if the angle at the centre of the circle is θ, the total area is the area of the sector =$\mathrm{=(\frac{θ}{360°})\times πr^2}$, where θ is the central angle, in degrees, r is the radius of the circle.

• In different words, πr² represents the area of a complete circle, and $\mathrm{\frac{θ}{360°}}$ suggests how a lot of the circle is covered in the sector.

Sector formula

If the angle at the core(centre) is θ in radians then

$$\mathrm{The\: area\: of\: the\: sector = \frac{1}{2} × r^2 θ,}$$

where θ is the angle made at the middle of the contrary(opposite) side in θ radians, r is the radius of the circle.

Note that semicircles and quadrants are exceptionally formed sectors with angles of 180° and 90°.

Solved Examples

1)What is the area of the sector of a circle with radius 6 and the angle being 60°?

Ans. The area of the sector$\mathrm{=(\frac{θ}{360°}) × πr^2=(\frac{60°}{360°}) × π.6^2=6π}$

Hence, the area is 6π.

2)What is the area of the sector of a circle with radius 5 and the angle being 120°?

Ans. The area of the sector$\mathrm{=(\frac{θ}{360°}) × πr^2=(\frac{120°}{360°}) × π.5^2=8.3π}$

Hence, the area is 8.3π.

Conclusion

In this tutorial, we learned about the sectors of a circle and the area of a sector of a circle. The sector indicates the area of a portion of a circle. We know that the area of a circle is calculated by the use of the formula πr². We know that an ideal circle is 360 degrees. The area of a circle with an angle of 360° at its middle is given by way of πr², where r is the radius of the circle. If the angle at the core of the circle is 1°, the area is $\mathrm{\frac{πr^2}{360°}}$. So, if the angle subtended at the center of the circle is θ, the area of sector =$\mathrm{(\frac{θ}{360°})\times πr^2}$, where θ is the angle subtended at the center of the circle, expressed in degrees, r is the radius of the circle. We also solved some examples using the formula of area of sector.

FAQs

1. What is the area of a sector of a circle?

The space enclosed by using the sector-shaped circle is called the area of the sector. The component of a circle bounded by means of two radii and associated arcs is known as a sector.

2. What is the method for the area of a sector of a circle?

The two major formulas used to locate the area of a sector are: , the area 'r' is the radius of the circle.

$$\mathrm{Circle\: sector\: area =\frac{1}{2} × r^2 θ,}$$

where θ is the angle from the middle in radians and 'r' is the radius of the circle.

3. What is meant by a sector of a circle?

A sector is defined as the element of a circle bounded between two radii and adjoining arcs. A semicircle is the most common sector of a circle and represents one $\mathrm{\frac{1}{2}}$ of a circle.

4. What does arc mean?

A portion of a curve or part of a circle is called an arc. Many objects have curves in their shape. The curved components of these objects are mathematically called arcs.

5. How can I locate the area of a sector given the length and radius of an arc?

Given the size and radius of an arc, you can calculate the area of a sector. First, calculate the arc angle (θ) using the formula arc length $\mathrm{=(\frac{θ}{360° }) × πr^2}$. Now you recognize the radius. Knowing the angle, you can calculate the area of the sector the use of the following formula −

$$\mathrm{Circle\: sector\: area =(\frac{θ}{360° }) × πr^2}$$

6. How to discover the radius from the area of the sector?

You can calculate the radius through substituting the values into the formula −

$$\mathrm{area\: of\: a\: circle =(\frac{θ}{360° }) × πr^2}$$

Substituting the given values into the formula, area $\mathrm{= (\frac{θ}{360° }) × πr^2}$, i.e. $\mathrm{H.36π = (\frac{90}{360} × πr^2}$ which gives a value of r²=144 which means r = 12 units.

7. How can I locate the area of an area in terms of pi?

Sector area can also be expressed in the form of pi (π). For example, if the radius of the circle is four units and the arc angle of the sector is 90°, let's calculate the area of the sector using pi. Sector area = $\mathrm{=(\frac{θ}{360°}) × πr^2 }$. Substituting the values into the equation, the area of the sector $\mathrm{= (\frac{90}{360}) × π×42 }$ Solving this gives an area to be 4π.

8. How do I find the area of a sector in radians?

To find the area of a sector with a central angle in radians, use the formula

$$\mathrm{Area =\frac{1}{2} × r^2 θ, }$$

the angle θ is the middle angle in radians and 'r' is the radius of the circle.