{"id":1325,"date":"2023-05-04T10:00:18","date_gmt":"2023-05-04T04:30:18","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=1325"},"modified":"2023-05-05T09:17:43","modified_gmt":"2023-05-05T03:47:43","slug":"area-of-a-sector-of-a-circle","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/","title":{"rendered":"How To Find The Area Of A Sector Of A Circle"},"content":{"rendered":"<h2><strong>How To Find The Area Of A Sector Of A Circle<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12082\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg\" alt=\"\" width=\"869\" height=\"428\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg 869w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2-300x148.jpeg 300w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2-768x378.jpeg 768w\" sizes=\"(max-width: 869px) 100vw, 869px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-134745 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-1.png\" alt=\"How To Find The Area Of A Sector Of A Circle 1\" width=\"178\" height=\"201\" \/><br \/>\nIf the arc subtends an angle of \u03b8 at the centre, then its arc length is<br \/>\n\\( \\frac{\\text{ }\\!\\!\\theta\\!\\!\\text{ }}{\\text{180}}\\text{ }\\!\\!\\times\\!\\!\\text{ }\\!\\!\\pi\\!\\!\\text{ r} \\)<br \/>\nHence, the arc length &#8216;l&#8217; of a sector of angle \u03b8 in a<br \/>\ncircle of radius r is given by<br \/>\n\\( l=\\frac{\\text{ }\\!\\!\\theta\\!\\!\\text{ }}{\\text{180}}\\text{ }\\!\\!\\times\\!\\!\\text{ }\\!\\!\\pi\\!\\!\\text{ r }&#8230;&#8230;.\\text{ (i)} \\)<br \/>\nIf the arc subtends an angle \u03b8, then area of the corresponding sector is<br \/>\n\\( \\frac{\\pi {{r}^{2}}\\theta }{360} \\)<br \/>\nThus, the area A of a sector of angle \u03b8 in a circle of radius r is given by<br \/>\n\\( A=~\\frac{\\theta }{360}\\text{ }\\times \\text{ }\\!\\!\\pi\\!\\!\\text{ }{{r}^{2}} \\)<br \/>\n\\( =\\frac{\\theta }{360}\\text{ }\\times \\text{ }\\left( \\text{Area of the circle} \\right)\\text{ }&#8230;&#8230;..\\text{ (ii)} \\)<br \/>\n= \u00d7 (Area of the circle) &#8230;.(ii)<br \/>\nSome useful results to remember:<br \/>\n(i) Angle described by minute hand in 60 minutes = 360\u00ba<br \/>\nAngle described by minute hand in one minute<br \/>\n\\( ={{\\left( \\frac{360}{60} \\right)}^{0}}=\\text{ }6{}^\\text{o} \\)<br \/>\nThus, minute hand rotates through an angle of 6\u00ba in one minute.<br \/>\n(ii) Angle described by hour hand in 12 hours = 360\u00ba<br \/>\nAngle described by hour hand in one hour<br \/>\n\\( =\\left( \\frac{360}{12} \\right)_{{}}^{0}=30{}^\\text{o} \\)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12174 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector.jpg\" alt=\"\" width=\"728\" height=\"546\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector.jpg 728w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector-300x225.jpg 300w\" sizes=\"(max-width: 728px) 100vw, 728px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12175\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector-1.png\" alt=\"\" width=\"846\" height=\"406\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector-1.png 846w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector-1-300x144.png 300w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-sector-1-768x369.png 768w\" sizes=\"(max-width: 846px) 100vw, 846px\" \/><\/p>\n<p><strong>Read More:<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/www.aplustopper.com\/parts-of-the-circle\/\"><span style=\"font-weight: 400;\">Parts of a Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/perimeter-of-a-circle\/\"><span style=\"font-weight: 400;\">Perimeter of A Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/common-chord-two-intersecting-circles\/\"><span style=\"font-weight: 400;\">Common Chord of Two Intersecting Circles<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/construction-of-a-circle\/\"><span style=\"font-weight: 400;\">Construction of a Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/area-of-a-circle\/\"><span style=\"font-weight: 400;\">The Area of A Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/properties-of-circles\/\"><span style=\"font-weight: 400;\">Properties of Circles<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/sector-of-a-circle\/\"><span style=\"font-weight: 400;\">Sector of A Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/area-of-a-segment-of-a-circle\/\"><span style=\"font-weight: 400;\">The Area of A Segment of A Circle<\/span><\/a><\/li>\n<\/ul>\n<h2><strong>Area Of A Sector Of A Circle With Examples<\/strong><\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/D9xX8ip7kRc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Example 1:<\/strong> \u00a0 \u00a0A sector is cut from a circle of radius 21 cm. The angle of the sector is 150\u00ba. Find the length of its arc and area.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 The arc length l and area A of a sector of angle \u03b8 in a circle of radius r are given by<br \/>\n\\( l=\\frac{\\theta }{360}\\times 2\\pi r\\text{\u00a0 and\u00a0 A=}\\frac{\\theta }{360}\\times \\pi {{r}^{2}}\\text{ respectively}\\text{.} \\)<br \/>\nHere, r = 21 cm and q = 150<br \/>\n\\(\\text{ }l=\\left\\{ \\frac{150}{360}\\times 2\\times \\frac{22}{7}\\times 21 \\right\\}\\text{cm}=\\text{55 cm} \\)<br \/>\n\\( \\text{and A}=\\left\\{ \\frac{150}{360}\\times \\frac{22}{7}\\times {{(21)}^{2}} \\right\\}c{{m}^{2}}=\\frac{1155}{2}c{{m}^{2}} \\)<br \/>\n= 577.5 cm<sup>2<\/sup><\/p>\n<p><strong>Example 2:<\/strong> \u00a0 \u00a0Find the area of the sector of a circle whose radius is 14 cm and angle of sector is 45\u00ba.<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong> We know that the area A of a sector of angle \u03b8 in a circle of radius r is given by<br \/>\n\\( A=\\frac{\\theta }{360}\\text{ }\\times \\text{ }\\pi {{r}^{2}} \\)<br \/>\nHere, r = 14 cm and \u03b8 = 45<br \/>\n\\(A=\\left\\{ \\frac{45}{360}\\times \\frac{22}{7}\\times {{(14)}^{2}} \\right\\}c{{m}^{2}}\\)<br \/>\n\\( =\\left\\{ \\frac{1}{8}\\times \\frac{22}{7}\\times 14\\times 14 \\right\\}c{{m}^{2}} \\)<br \/>\n= 77 cm<sup>2<\/sup><\/p>\n<p><strong>Example 3:<\/strong> \u00a0 \u00a0In Fig. there are shown sectors of two concentric circles of radii 7 cm and 3.5 cm. Find the area of the shaded region. (Use \u03c0 = \\(\\frac { 22 }{ 7 }\\)).<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong> Let A<sub>1<\/sub> and A<sub>2<\/sub> be the areas of sectors OAB and OCD respectively. Then, A<sub>1<\/sub> = Area of a sector of angle 30\u00ba in a circle of radius 7 cm<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-134752\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-2.png\" alt=\"How To Find The Area Of A Sector Of A Circle 2\" width=\"271\" height=\"139\" \/><br \/>\n\\( {{A}_{1}}=\\left\\{ \\frac{30}{360}\\times \\frac{22}{7}\\times {{7}^{2}} \\right\\}c{{m}^{2}}\\text{ }\\left[ U\\sin g:A=\\frac{\\theta }{360}\\times \\pi {{r}^{2}}\\, \\right] \\)<br \/>\n\u21d2 A<sub>1<\/sub> = \\(\\frac { 77 }{ 6 }\\) cm<sup>2<\/sup><br \/>\nA<sub>2<\/sub> = Area of a sector of angle 30\u00ba in a circle of radius 3.5 cm.<br \/>\n\u2234 \u00a0Area of the shaded region<br \/>\n\\({{A}_{2}}=\\left\\{ \\frac{30}{360}\\times \\frac{22}{7}\\times {{(3.5)}^{2}} \\right\\}c{{m}^{2}} \\)<br \/>\n\\({{A}_{2}}=\\left\\{ \\frac{1}{12}\\times \\frac{22}{7}\\times \\frac{7}{2}\\times \\frac{7}{2} \\right\\}c{{m}^{2}}=\\frac{77}{24}c{{m}^{2}}\u00a0\\)<br \/>\n\\( ={{A}_{1}}-{{A}_{2}}=\\left( \\frac{77}{6}-\\frac{77}{24} \\right) \\)<br \/>\n=\u00a0\\(\\frac { 77 }{ 24 }\\) \u00d7 (4 \u2013 1) cm<sup>2<\/sup> = \\(\\frac { 77 }{ 8 }\\) cm<sup>2<\/sup> = 9.625 cm<sup>2<\/sup><\/p>\n<p><strong>Example 4:<\/strong> \u00a0 \u00a0A pendulum swings through an angle of 30\u00ba and describes an arc 8.8 cm in length. Find the length of the pendulum.<br \/>\n<strong>Sol.<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0 Here, q = 30\u00ba, <em>l<\/em> = arc = 8.8 cm<br \/>\n\\( l=\\frac{\\theta }{360}\\times 2\\pi r \\)<br \/>\n\\( 8.8=~\\frac{30}{360}\\times 2\\times \\frac{22}{7}\\times r~ \\)<br \/>\n\\( r=\\frac{8.8\\times 6\\times 7}{22}=16.8\\text{ }cm \\)<\/p>\n<p><strong>Example 5:<\/strong> \u00a0 \u00a0The length of minute hand of a clock is 14 cm. Find the area swept by the minute hand in one minute. (Use \u03c0 = 22\/7)<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 \u00a0Clearly, minute hand of a clock describes a circle of radius equal to its length i.e., 14 cm.<br \/>\nSince the minute hand rotates through 6\u00ba in one minute. Therefore, area swept by the minute hand in one minute is the area of a sector of angle 6\u00ba in a circle of radius 14 cm. Hence, required area A is given by<br \/>\n\\( A=\\frac{\\theta }{360}\\times \\pi {{r}^{2}}~ \\)<br \/>\n\\( A=\\left\\{ \\frac{6}{360}\\times \\frac{22}{7}\\times {{(14)}^{2}} \\right\\} \\)<br \/>\n\\( A=\\left\\{ \\frac{1}{60}\\times \\frac{22}{7}\\times 14\\times 14 \\right\\}=\\frac{154}{15} \\)<br \/>\n= 10.26 cm<sup>2<\/sup><\/p>\n<p><strong>Example 6:<\/strong> \u00a0 \u00a0The perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm. Find the area of the sector.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let OAB be the given sector. Then,<br \/>\nPerimeter of sector OAB = 16.4 cm<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-134753\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-3.png\" alt=\"How To Find The Area Of A Sector Of A Circle 3\" width=\"171\" height=\"168\" \/><br \/>\n\u21d2 OA + OB + arc AB = 16.4 cm<br \/>\n\u21d2 5.2 + 5.2 + arc AB = 16.4<br \/>\n\u21d2 arc AB = 6 cm<br \/>\n\u21d2 <em>l<\/em> = 6 cm<br \/>\n\u2234\u00a0Area of\u00a0 sector OAB = \\(\\frac { 1 }{ 2 }\\)\u00a0<em>l<\/em>r<br \/>\n= \\(\\frac { 1 }{ 2 }\\) \u00d7 6 \u00d7 5.2 cm<sup>2<\/sup> = 15.6 cm<sup>2<\/sup><\/p>\n<p><strong>Example 7:<\/strong> \u00a0 \u00a0The minute hand of a clock is 10cm long. Find the area of the face of the clock described by the minute hand between 9 A.M. and 9.35 A.M.<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong> We have,<br \/>\nAngle described by the minute hand in one minute = 6\u00ba<br \/>\n\u2234 Angle described by the minute hand in<br \/>\n35 minutes = (6 \u00d7 35)\u00ba = 210\u00ba<br \/>\n\u2234 Area swept by the minute hand in 35 minutes<br \/>\n= Area of a sector of angle 210\u00ba in a circle of radius 10 cm<br \/>\n\\(=\\left\\{ \\frac{210}{360}\\times \\frac{22}{7}\\times {{(10)}^{2}} \\right\\}c{{m}^{2~}}=\\text{ }183.3\\text{ }c{{m}^{2}}~\\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 }\\left[ \\text{Using \u00a0}A=\\frac{\\theta }{360{}^\\text{o}}\\times \\pi {{r}^{2}} \\right] \\)<\/p>\n<p><strong>Example 8:<\/strong> \u00a0 \u00a0The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 2 days.<br \/>\n(Take \u03c0 = 22\/7)<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 In 2 days, the short hand will complete 4 rounds.<br \/>\n\u2234 Distance moved by its tip = 4 (Circumference of a circle of radius 4 cm)<br \/>\n\\( =4\\times \\left( 2\\times \\frac{22}{7}\\times 4 \\right)=\\frac{704}{7}\\text{ cm} \\)<br \/>\nIn 2 days, the long hand will complete 48 rounds.<br \/>\n\u2234 Distance moved by its tip<br \/>\n= 48 (Circumference of a circle of radius 6 cm)<br \/>\n\\( =48\\times \\left( 2\\times \\frac{22}{7}\\times 4 \\right)=\\frac{12672}{7}\\text{ cm} \\)<br \/>\nHence,<br \/>\nSum of the distance moved by the tips of two hands of the clock<br \/>\n\\( =\\left( \\frac{704}{7}+\\frac{12672}{7} \\right)=\\text{ }1910.57\\text{ }cm \\)<\/p>\n<p><strong>Example 9:<\/strong> \u00a0 \u00a0An elastic belt is placed round the rim of a pulley of radius 5 cm. One point on the belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from O. Find the length of the belt that is in contact with the rim of the pulley. Also, find the shaded area.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 In the adjacent figure, let \u2220AOP = \u2220BOP = \u03b8. Clearly, portion AB of the belt is not in contact with the rim of the pulley. In right triangle OAP, we have<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-134754\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-4.png\" alt=\"How To Find The Area Of A Sector Of A Circle 4\" width=\"343\" height=\"236\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-4.png 343w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-4-300x206.png 300w\" sizes=\"(max-width: 343px) 100vw, 343px\" \/><br \/>\n\\( \\cos \\theta =\\frac{OA}{OP}=\\frac{5}{10}=\\frac{1}{2}\\text{ }\\Rightarrow \\text{ }\\theta =\\text{ }60{}^\\text{o} \\)<br \/>\n\u21d2 \u2220AOB = 2\u03b8 = 120\u00ba<br \/>\n\\(\\text{Arc AB}=\\frac{120{}^\\text{o}\\times 2\\times \\pi \\times 5}{360}=\\frac{10\\pi }{3}cm \\)<br \/>\n\\( \\left[ \\text{Using }l=\\frac{\\theta }{360}\\times 2\\pi r \\right] \\)<br \/>\nHence, Length of the belt that is in contact with the rim of the pulley<br \/>\n= Circumference of the rim \u2013 Length of arc AB<br \/>\n\\( =\\text{ }2\\pi \\times 5-\\frac{10\\pi }{3}=\\frac{20\\pi }{3}~cm \\)<br \/>\nNow,<br \/>\nArea of sector OAQB =\u00a0\\(\\frac { 1 }{ 2 }\\)\u00a0\u00d7 \u03c0\u00a0\u00d7 5<sup>2<\/sup>\u00a0cm<sup>2<\/sup><br \/>\n\\( =\\frac{25\\pi }{3}\\text{ c}{{\\text{m}}^{2}}\\text{\u00a0\u00a0 }\\left[ \\text{Using}Area=\\frac{\\theta }{360}\\times \\pi {{r}^{2}} \\right] \\)<br \/>\nArea of quadrilateral OAPB = 2 (Area of \u2206OAP)<br \/>\n= 2 \u00d7 (1\/2 \u00d7 OA \u00d7 AP)<br \/>\n= 5 \u00d7 5\u221a3 cm<sup>2<\/sup><br \/>\n[\u2235 OP<sup>2<\/sup> = OA<sup>2<\/sup> + AP<sup>2 <\/sup>\u00a0\u00a0\u21d2 \u00a0AP =\u00a0\\(\\sqrt{100-25}\\) = 5\u221a3 ]<br \/>\n= 25\u221a3 cm<sup>2<\/sup><br \/>\nHence,<br \/>\nShaded area = Area of quadrilateral<br \/>\nOAPB \u2013 Area of sector OAQB.<br \/>\n\\( =\\left( 25\\sqrt{3}-\\frac{25\\pi }{3} \\right)=\\frac{25}{3}(3\\sqrt{3}-\\pi )\\text{ c}{{\\text{m}}^{\\text{2}}} \\)<\/p>\n<p><strong>Example 10:<\/strong> \u00a0 \u00a0An arc of a circle is of length 5\u03c0 cm and the sector it bounds has an area of 20 \u03c0 cm\u00b2. Find the radius of the circle.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let the radius of the circle be r cm and the arc AB of length 5\u03c0 cm subtends angle \u03b8 at the centre O of the circle. Then,<br \/>\nArc AB = 5\u03c0 cm and<br \/>\nArea of sector OAB = 20\u03c0 cm<sup>2<\/sup><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-134755\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-5.png\" alt=\"How To Find The Area Of A Sector Of A Circle 5\" width=\"283\" height=\"220\" \/><br \/>\n\\( \\Rightarrow \\frac{\\theta }{360}\\times 2\\pi r=5\\pi \\text{ and }\\frac{\\theta }{360}\\times \\pi {{r}^{2}}=20\\pi \\)<br \/>\n\\( \\Rightarrow \\frac{\\frac{\\theta }{360}\\times \\pi {{r}^{2}}}{\\frac{\\theta }{360}\\times 2\\pi r}=\\frac{20\\pi }{5\\pi } \\)<br \/>\n\u21d2 r\/2 = 4 \u00a0\u21d2 \u00a0 r = 8 cm<br \/>\n<strong>ALTER:<\/strong> We have, Area = 1\/2 <em>l\u00a0<\/em>r \u21d220\u03c0<br \/>\n= 1\/2 \u00d7 5\u03c0 \u00d7 r = 8 cm<\/p>\n<p><strong>Example 11: \u00a0<\/strong> \u00a0An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm. Find the area between the two consecutive ribs of the umbrella.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Since ribs are equally spaced. Therefore,<br \/>\nAngle made by two consecutive ribs at the centre = \\(\\frac { 360 }{ 8 }\\) = 45\u00ba<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-134756\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-6.png\" alt=\"How To Find The Area Of A Sector Of A Circle 6\" width=\"351\" height=\"226\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-6.png 351w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-6-300x193.png 300w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><br \/>\nThus,<br \/>\nArea between two consecutive ribs<br \/>\n= Area of a sector of a circle of radius 45 cm and sector angle 45\u00ba<br \/>\n\\(=\\left\\{ \\frac{45}{360}\\times \\frac{22}{7}\\times 45\\times 45 \\right\\}\\text{c}{{\\text{m}}^{2}}\\text{ }\\left[ \\text{Using }Area=\\frac{\\theta }{360}\\times \\pi {{r}^{2}} \\right] \\)<br \/>\n=\u00a0 \\(\\frac { 1 }{ 8 }\\)\u00a0\u00d7 \\(\\frac { 22 }{ 7 }\\)\u00a0\u00d7 45\u00d7 45 cm<sup>2<\/sup> = 795.53 cm<sup>2<\/sup><\/p>\n<p><strong>Example 12: \u00a0<\/strong> \u00a0A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. Find:<br \/>\n(i) the total length of the silver wire required<br \/>\n(ii) the area of each sector of the brooch.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 <strong>(i)<\/strong> We have,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-134757\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-7.png\" alt=\"How To Find The Area Of A Sector Of A Circle 7\" width=\"226\" height=\"225\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-7.png 226w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-7-150x150.png 150w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Find-The-Area-Of-A-Sector-Of-A-Circle-7-100x100.png 100w\" sizes=\"(max-width: 226px) 100vw, 226px\" \/><br \/>\nTotal length of the silver wire = Circumference of the circle of radius<br \/>\n35\/2 mm + Length of five diameters<br \/>\n=\u00a02\u03c0 \u00d7 \\(\\frac { 35 }{ 2 }\\) + 5 \u00d7 35 mm<br \/>\n\\( =\\left( 2\\times \\frac{22}{7}\\times \\frac{35}{2}+175 \\right)\\text{ mm} \\)<br \/>\n= 285 mm<br \/>\n<strong>(ii)<\/strong>\u00a0 The circle is divided into 10 equal sectors, Therefore, Area of each sector of the brooch<br \/>\n= 1\/10 (Area of the circle)<br \/>\n= 1\/10 \u00d7 \u03c0 \u00a0\u00d7 (35\/2)<sup>2<\/sup>\u00a0cm<sup>2<\/sup><br \/>\n\\( \\text{=}\\frac{1}{10}\\times \\frac{22}{7}\\times \\frac{35}{2}\\times \\frac{35}{2}\\text{ m}{{\\text{m}}^{2}} \\)<br \/>\n\\( =\\frac{385}{4}\\text{ m}{{\\text{m}}^{2}} \\)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>How To Find The Area Of A Sector Of A Circle If the arc subtends an angle of \u03b8 at the centre, then its arc length is Hence, the arc length &#8216;l&#8217; of a sector of angle \u03b8 in a circle of radius r is given by If the arc subtends an angle \u03b8, then [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[287,286,278,284,285],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>How To Find The Area Of A Sector Of A Circle - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How To Find The Area Of A Sector Of A Circle\" \/>\n<meta property=\"og:description\" content=\"How To Find The Area Of A Sector Of A Circle If the arc subtends an angle of \u03b8 at the centre, then its arc length is Hence, the arc length &#8216;l&#8217; of a sector of angle \u03b8 in a circle of radius r is given by If the arc subtends an angle \u03b8, then [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/\" \/>\n<meta property=\"og:site_name\" content=\"A Plus Topper\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/aplustopper\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-05-04T04:30:18+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-05-05T03:47:43+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Veerendra\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.aplustopper.com\/#organization\",\"name\":\"Aplus Topper\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"sameAs\":[\"https:\/\/www.facebook.com\/aplustopper\/\"],\"logo\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#logo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"width\":1585,\"height\":375,\"caption\":\"Aplus Topper\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/#logo\"}},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.aplustopper.com\/#website\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"name\":\"A Plus Topper\",\"description\":\"Improve your Grades\",\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.aplustopper.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg\",\"width\":869,\"height\":428},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#webpage\",\"url\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/\",\"name\":\"How To Find The Area Of A Sector Of A Circle - A Plus Topper\",\"isPartOf\":{\"@id\":\"https:\/\/www.aplustopper.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#primaryimage\"},\"datePublished\":\"2023-05-04T04:30:18+00:00\",\"dateModified\":\"2023-05-05T03:47:43+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.aplustopper.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"How To Find The Area Of A Sector Of A Circle\"}]},{\"@type\":\"Article\",\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#webpage\"},\"author\":{\"@id\":\"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e\"},\"headline\":\"How To Find The Area Of A Sector Of A Circle\",\"datePublished\":\"2023-05-04T04:30:18+00:00\",\"dateModified\":\"2023-05-05T03:47:43+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#webpage\"},\"wordCount\":1656,\"commentCount\":1,\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg\",\"keywords\":[\"Area of a Sector\",\"Area Of A Sector Of A Circle\",\"Circles\",\"Sector\",\"Sector Of A Circle\"],\"articleSection\":[\"Mathematics\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#respond\"]}]},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e\",\"name\":\"Veerendra\",\"image\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#personlogo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g\",\"caption\":\"Veerendra\"},\"url\":\"https:\/\/www.aplustopper.com\/author\/veerendra\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"How To Find The Area Of A Sector Of A Circle - A Plus Topper","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/","og_locale":"en_US","og_type":"article","og_title":"How To Find The Area Of A Sector Of A Circle","og_description":"How To Find The Area Of A Sector Of A Circle If the arc subtends an angle of \u03b8 at the centre, then its arc length is Hence, the arc length &#8216;l&#8217; of a sector of angle \u03b8 in a circle of radius r is given by If the arc subtends an angle \u03b8, then [&hellip;]","og_url":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/","og_site_name":"A Plus Topper","article_publisher":"https:\/\/www.facebook.com\/aplustopper\/","article_published_time":"2023-05-04T04:30:18+00:00","article_modified_time":"2023-05-05T03:47:43+00:00","og_image":[{"url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg"}],"twitter_card":"summary","twitter_misc":{"Written by":"Veerendra","Est. reading time":"8 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/www.aplustopper.com\/#organization","name":"Aplus Topper","url":"https:\/\/www.aplustopper.com\/","sameAs":["https:\/\/www.facebook.com\/aplustopper\/"],"logo":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#logo","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","width":1585,"height":375,"caption":"Aplus Topper"},"image":{"@id":"https:\/\/www.aplustopper.com\/#logo"}},{"@type":"WebSite","@id":"https:\/\/www.aplustopper.com\/#website","url":"https:\/\/www.aplustopper.com\/","name":"A Plus Topper","description":"Improve your Grades","publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.aplustopper.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#primaryimage","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg","width":869,"height":428},{"@type":"WebPage","@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#webpage","url":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/","name":"How To Find The Area Of A Sector Of A Circle - A Plus Topper","isPartOf":{"@id":"https:\/\/www.aplustopper.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#primaryimage"},"datePublished":"2023-05-04T04:30:18+00:00","dateModified":"2023-05-05T03:47:43+00:00","breadcrumb":{"@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.aplustopper.com\/"},{"@type":"ListItem","position":2,"name":"How To Find The Area Of A Sector Of A Circle"}]},{"@type":"Article","@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#article","isPartOf":{"@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#webpage"},"author":{"@id":"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e"},"headline":"How To Find The Area Of A Sector Of A Circle","datePublished":"2023-05-04T04:30:18+00:00","dateModified":"2023-05-05T03:47:43+00:00","mainEntityOfPage":{"@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#webpage"},"wordCount":1656,"commentCount":1,"publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"image":{"@id":"https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#primaryimage"},"thumbnailUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg","keywords":["Area of a Sector","Area Of A Sector Of A Circle","Circles","Sector","Sector Of A Circle"],"articleSection":["Mathematics"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.aplustopper.com\/area-of-a-sector-of-a-circle\/#respond"]}]},{"@type":"Person","@id":"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e","name":"Veerendra","image":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g","caption":"Veerendra"},"url":"https:\/\/www.aplustopper.com\/author\/veerendra\/"}]}},"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/1325"}],"collection":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/comments?post=1325"}],"version-history":[{"count":1,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/1325\/revisions"}],"predecessor-version":[{"id":159319,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/1325\/revisions\/159319"}],"wp:attachment":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/media?parent=1325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/categories?post=1325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/tags?post=1325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}