The sector has central angle $\theta$ and radius $r$. If angle $\theta$ in degrees, \begin{align} & \boxed{ \text{Sector area} = {\theta \over 360^\circ} \times \pi r^2 } \\ \\ & \boxed{ \text{Arc length} = {\theta \over 360^\circ} \times 2 \pi r} \end{align} Show If angle $\theta$ in radians, \begin{align} & \boxed{ \text{Sector area} = {1 \over 2} r^2 \theta } \\ \\ & \boxed{ \text{Arc length} = r \theta } \end{align}
SlicesThere are two main "slices" of a circle:
Try Them!Common SectorsThe Quadrant and Semicircle are two special types of Sector: Half a circle is a Semicircle. Quarter of a circle is a Quadrant. You can work out the Area of a Sector by comparing its angle to the angle of a full circle. Note: we are using radians for the angles. This is the reasoning:
A circle has an angle of 2π and an Area of:πr2 A Sector has an angle of θ instead of 2π so its Area is : θ2π × πr2 Which can be simplified to:θ2 × r2 Area of Sector = θ 2 × r2 (when θ is in radians) Area of Sector = θ × π 360 × r2 (when θ is in degrees) Area of SegmentThe Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here). There is a lengthy reason, but the result is a slight modification of the Sector formula: Area of Segment = θ − sin(θ) 2 × r2 (when θ is in radians) Area of Segment = ( θ × π 360 − sin(θ)2 ) × r2 (when θ is in degrees) Arc LengthThe arc length (of a Sector or Segment) is: L = θ × r (when θ is in radians) L = θ × π180 × r (when θ is in degrees) Copyright © 2017 MathsIsFun.com
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. Table of ContentsCircle and its Different ParametersA 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. RadiusThe radius\(\left( r \right)\) of a circle is a line segment joining the centre and any point on the circumference. DiameterThe 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. ArcAny 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. SectorThe 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 CircleThe 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 CircleIf \(r\) is the radius of a circle, then the area of a circle is given by \(\pi {r^2}.\) Area of SectorThe 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, 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 SectorThe 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 SectorThe 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 SectorQ.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, \(\pi = 3.14.\) 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 \(π=3.14.\) 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 \(π=3.14.\) 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 \(π=3.14.\) 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 \(\pi = \frac{{22}}{7}.\) Therefore, the area between the two consecutive ribs of the umbrella is \(77\,{\rm{uni}}{{\rm{t}}^2}.\) SummaryA 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 SectorQ.1. What is the area of the major sector? Q.2. What is the formula for the area of a sector? Q.3. What is a perimeter of a sector? Q.4. What is a sector in circles? Q.5. What is the area of the minor sector? 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. Happy Embibing! |