{"id":1289,"date":"2020-12-07T09:59:06","date_gmt":"2020-12-07T04:29:06","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=1289"},"modified":"2020-12-07T16:34:55","modified_gmt":"2020-12-07T11:04:55","slug":"area-of-a-circle","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/area-of-a-circle\/","title":{"rendered":"How To Calculate The Area Of A Circle"},"content":{"rendered":"<h2>How To Calculate The Area Of A Circle<\/h2>\n<p>A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point always remains same.<br \/>\nThe fixed point is called the centre and the given constant distance is known as the radius of the circle.<br \/>\nThe perimeter of a circle is known as its circumference.<br \/>\nIf r is the radius of a circle, then<br \/>\n(i) Circumference = 2\u03c0r or \u03c0d, where d = 2r is the diameter of the circle.<br \/>\n(ii) Area = \u03c0r<sup>2<\/sup> or \u03c0d<sup>2<\/sup>\/4<br \/>\n(iii) Area of semi-circle = \u03c0r<sup>2<\/sup>\/2<br \/>\n(iv) Area of a quadrant of a circle =\u00a0\u03c0r<sup>2<\/sup>\/4<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-12082\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg\" alt=\"\" width=\"869\" height=\"428\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2.jpeg 869w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2-300x148.jpeg 300w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-2-768x378.jpeg 768w\" sizes=\"(max-width: 869px) 100vw, 869px\" \/><\/p>\n<p><iframe loading=\"lazy\" title=\"Area of a circle\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/ghSBNOt-b-U?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Read More:<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/cbselibrary.com\/parts-of-the-circle\/\"><span style=\"font-weight: 400;\">Parts of a Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/perimeter-of-a-circle\/\"><span style=\"font-weight: 400;\">Perimeter of A Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.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:\/\/cbselibrary.com\/construction-of-a-circle\/\"><span style=\"font-weight: 400;\">Construction of a Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/properties-of-circles\/\"><span style=\"font-weight: 400;\">Properties of Circles<\/span><\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/sector-of-a-circle\/\"><span style=\"font-weight: 400;\">Sector of A Circle<\/span><\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.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<li><a href=\"https:\/\/cbselibrary.com\/area-of-a-sector-of-a-circle\/\"><span style=\"font-weight: 400;\">The Area of A Sector of A Circle<\/span><\/a><\/li>\n<\/ul>\n<h2><strong>Area Enclosed By Two Concentric Circles<\/strong><\/h2>\n<p>If R and r are radii of two con-centric circles, then<br \/>\narea enclosed by the two circles = \u03c0R<sup>2<\/sup> \u2013 \u03c0r<sup>2<\/sup> = \u03c0(R<sup>2<\/sup> \u2013 r<sup>2<\/sup>) = \u03c0(R + r) (R \u2013 r)<\/p>\n<p><img loading=\"lazy\" class=\"wp-image-12084 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/area-of-a-circle-1.png\" alt=\"\" width=\"326\" height=\"359\" \/><\/p>\n<p><iframe loading=\"lazy\" title=\"Area of Shaded Region Concentric Circles\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/m3dwUx5pXj8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Some useful results:<\/strong><br \/>\n(i) If two circles touch internally, then the distance between their centres is equal to the difference of their radii.<br \/>\n(ii) If two circles touch externally, then the distance between their centres is equal to the sum of their radii.<br \/>\n(iii) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.<br \/>\n(iv) The number of revolutions completed by a rotating wheel in one minute<br \/>\n\\(\\frac{\\text{Distance moved in one minute}}{\\text{Circumference}} \\)<\/p>\n<h2>Area Of A Circle With Examples<\/h2>\n<p><strong>Example 1: \u00a0 \u00a0<\/strong>Find the area of a circle whose circumference is 22 cm.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let r be the radius of the circle. Then,<br \/>\nCircumference = 22 cm<br \/>\n\u21d2 2\u03c0r = 22<br \/>\n\u21d2 2 \u00d7 \\(\\frac { 22 }{ 7 }\\)\u00a0\u00d7 r = 22<br \/>\n\u21d2 r = \\(\\frac { 7 }{ 2 }\\)\u00a0cm<br \/>\ntherefore Area of the circle = \u03c0r\u00b2 =\u00a0\\(\\frac { 22 }{ 7 }\\)\u00a0\u00d7 7\u00b2<br \/>\n= 38.5 cm<sup>2<\/sup><strong><br \/>\n<\/strong><\/p>\n<p><strong>Example 2: \u00a0 \u00a0<\/strong>Find the area of a quadrant of a circle whose circumference is 22 cm.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let r be the radius of the circle. Then,<br \/>\nCircumference = 22 cm<br \/>\n\u21d2 \u00a02\u03c0r = 22<br \/>\n\u21d2 2 \u00d7 \\(\\frac { 22 }{ 7 }\\)\u00a0\u00d7 r = 22<br \/>\n\u21d2 r = \\(\\frac { 7 }{ 2 }\\)\u00a0cm<br \/>\n\u2234 Area of a quadrant = \\(\\frac { 1 }{ 4 }\\) \u03c0r<sup>2<\/sup><br \/>\n= \u00a0\\(\\frac { 77 }{ 4 }\\) = 9.625 cm<sup>2<\/sup><\/p>\n<p><strong>Example 3: \u00a0 \u00a0<\/strong>Two circles touch externally. The sum of their areas is 130 sq. cm. and the distance between their centres is 14 cm. Find the radii of the circles.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 If two circles touch externally, then the distance between their centres is equal to the sum of their radii.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126558\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-1.png\" alt=\"How To Calculate The Area Of A Circle 1\" width=\"395\" height=\"269\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-1.png 395w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-1-300x204.png 300w\" sizes=\"(max-width: 395px) 100vw, 395px\" \/><br \/>\nLet the raddi of the two circles be r<sub>1<\/sub> cm and r<sub>2<\/sub> cm respectively.<br \/>\nLet C<sub>1<\/sub>\u00a0and C<sub>2<\/sub> be the centres of the given circles. Then,<br \/>\nC<sub>1<\/sub>C<sub>2<\/sub> = r<sub>1<\/sub> + r<sub>2<\/sub><br \/>\n\u21d2 14 = r<sub>1<\/sub> + r<sub>2<\/sub> \u00a0 \u00a0 \u00a0 [\u2235 C<sub>1<\/sub>C<sub>2<\/sub> = 14 cm (given)]<br \/>\n\u21d2 r<sub>1<\/sub> + r<sub>2<\/sub>\u00a0= 14 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8230;.(i)<br \/>\nIt is given that the sum of the areas of two circles is equal to 130 cm<sup>2<\/sup>.<br \/>\n\u2234 \u03c0r<sub>1<\/sub><sup>2<\/sup>+ \u03c0r<sub>2<\/sub><sup>2\u00a0<\/sup>= 130\u03c0<br \/>\n\u21d2 r<sub>1<\/sub><sup>2\u00a0<\/sup>r<sub>2<\/sub><sup>2<\/sup>\u00a0+ r<sub>2<\/sub><sup>2<\/sup>= 130 \u00a0 \u00a0 \u00a0 \u00a0 &#8230;.(ii)<br \/>\nNow,<br \/>\n(r<sub>1<\/sub> + r<sub>2<\/sub>)<sup>2<\/sup> = r<sub>1<\/sub><sup>2<\/sup> + r<sub>2<\/sub><sup>2<\/sup>\u00a0+ 2r<sub>1<\/sub>r<sub>2<\/sub><br \/>\n\u21d2 14<sup>2<\/sup> = 130 + 2r<sub>1<\/sub>r<sub>2 \u00a0 \u00a0 \u00a0 \u00a0<\/sub> [Using (i) and (ii)]<br \/>\n\u21d2 196 \u2013 130 = 2r<sub>1<\/sub>r<sub>2<\/sub><br \/>\n\u21d2\u00a0r<sub>1<\/sub> r<sub>2<\/sub> = 33 \u00a0 \u00a0 \u00a0 \u00a0&#8230;.(iii)<br \/>\nNow,<br \/>\n(r<sub>1<\/sub> \u2013 r<sub>2<\/sub>)<sup>2<\/sup> = r<sub>1<\/sub><sup>2<\/sup> + r<sub>2<\/sub><sup>2<\/sup> \u2013 2r<sub>1<\/sub>r<sub>2<\/sub><br \/>\n\u21d2 (r<sub>1<\/sub> \u2013 r<sub>2<\/sub>)<sup>2<\/sup> = 130 \u2013 2 \u00d7 33 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[Using (ii) and (iii)]<br \/>\n\u21d2 (r<sub>1<\/sub> \u2013 r<sub>2<\/sub>)<sup>2<\/sup> = 64 \u00a0 \u21d2 \u00a0\u00a0r<sub>1<\/sub> \u2013 r<sub>2<\/sub> = 8 \u00a0 \u00a0 \u00a0 \u00a0 &#8230;.(iv)<br \/>\nSolving (i) and (iv), we get r<sub>1<\/sub>\u00a0= 11 cm and r<sub>2<\/sub> = 3 cm.<br \/>\nHence, the radii of the two circles are 11 cm and 3 cm.<\/p>\n<p><strong>Example 4: \u00a0 \u00a0<\/strong>Two circles touch internally. The sum of their areas is 116 \u03c0 cm<sup>2<\/sup> and distance between their centres is 6 cm. Find the radii of the circles.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let R and r be the radii of the circles having centres at O and O&#8217; respectively. Then,<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126561\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-2a.png\" alt=\"How To Calculate The Area Of A Circle 2a\" width=\"189\" height=\"207\" \/><br \/>\nSum of areas = 116\u03c0 cm<sup>2<\/sup><br \/>\n\u21d2 \u03c0R<sup>2<\/sup> + \u03c0r<sup>2<\/sup> = 116 \u03c0<br \/>\n\u21d2 R<sup>2<\/sup> + r<sup>2<\/sup> = 116 &#8230;.(i)<br \/>\nDistance between the centres = 6 cm<br \/>\n\u21d2 OO&#8217; = 6 cm<br \/>\n\u21d2 R \u2013 r = 6 &#8230;.(ii)<br \/>\nNow, (R + r)<sup>2<\/sup> + (R \u2013 r)<sup>2<\/sup> = 2(R<sup>2<\/sup> + r<sup>2<\/sup>)<br \/>\n\u21d2 (R + r)<sup>2<\/sup> + 36 = 2 \u00d7 116 \u00a0 \u00a0 \u00a0 \u00a0[Using (i) and (ii)]<br \/>\n\u21d2 (R + r)<sup>2<\/sup> = (2 \u00d7 116 \u2013 36) = 196<br \/>\n\u21d2 R + r = 14 \u00a0 \u00a0 \u00a0 \u00a0 \u2026..(iii)<br \/>\nSolving (ii) and (iii), we get R = 10 and r = 4.<br \/>\nHence, radii of the given circles are 10 cm and 4 cm respectively.<\/p>\n<p><strong>Example 5: \u00a0 \u00a0<\/strong>A copper wire, when bent in the form of a square, encloses an area of 484 cm2. If the same wire is bent in the form of a circle, find the area enclosed by it (Use \u03c0 = \\(\\frac { 22 }{ 7 }\\) ).<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong> We have,<br \/>\nArea of the square = \u221a484 cm<sup>2<\/sup><br \/>\n\u2234 Side of the square \u221a484 cm = 22 cm \u00a0 \u00a0 \u00a0\u00a0\\( \\left[ \\because \\text{ }\\,\\,\\text{Area = (Side}{{\\text{)}}^{\\text{2}}}\\text{ }\\Rightarrow \\text{ Side = }\\sqrt{\\text{Area}} \\right] \\)<br \/>\nSo, Perimeter of the square = 4 (side)<br \/>\n= (4 \u00d7 22) cm = 88 cm<br \/>\nLet r be the radius of the circle. Then,<br \/>\nCircumference of the circle = Perimeter of the square.<br \/>\n\u21d2 2\u03c0r = 88<br \/>\n\u21d2 2 \u00d7 \\(\\frac { 22 }{ 7 }\\)\u00a0\u00d7 r = 88<br \/>\n\u21d2 r = 14 cm<br \/>\n\u2234 Area of the circle = \u03c0r<sup>2<\/sup><br \/>\n\\( \\text{=}\\left\\{ \\frac{\\text{22}}{\\text{7}}\\text{ }\\!\\!\\times\\!\\!\\text{ (14}{{\\text{)}}^{\\text{2}}} \\right\\}\\text{ c}{{\\text{m}}^{\\text{2}}} \\)<br \/>\n= 616 cm<sup>2<\/sup><\/p>\n<p><strong>Example 6: \u00a0 \u00a0<\/strong>Fig. depicts an archery target marked with its five scoring areas from the centre outwards as Gold, Red, Blue, Black and white. The diameter of the region representing Gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong> We have,<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126675\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-3.png\" alt=\"How To Calculate The Area Of A Circle 3\" width=\"213\" height=\"214\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-3.png 213w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-3-150x150.png 150w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2020\/12\/How-To-Calculate-The-Area-Of-A-Circle-3-100x100.png 100w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><br \/>\nr = Radius of the region representing Gold score = 10.5 cm<br \/>\n\u2234 r<sub>1<\/sub> = Radius of the region representing Gold and Red scoring areas<br \/>\n= (10.5 + 10.5) cm = 21 cm = 2r cm<br \/>\nr<sub>2<\/sub> = Radius of the region representing Gold, Red and Blue scoring areas<br \/>\n= (21 + 10.5) cm = 31.5 cm = 3r cm<br \/>\nr<sub>3<\/sub> = Radius of the region representing Gold, Red, Blue and Black scoring areas<br \/>\n= (31.5 + 10.5) cm = 42 cm = 4r cm<br \/>\nr<sub>4<\/sub> = Radius of the region representing Gold, Red, Blue, Black and white scoring areas<br \/>\n= (42 + 10.5) cm = 52.5 cm = 5r cm<br \/>\nNow, A<sub>1<\/sub> = Area of the region representing Gold scoring area<br \/>\n= \u03c0r<sup>2<\/sup> = 22\/7\u00d7 (10.5)<sup>2<\/sup> = 22\/7 \u00d7 10.5 \u00d7 10.5<br \/>\n= 22\u00a0\u00d7 1.5 \u00d7 10.5 = 346.5 cm<sup>2<\/sup><br \/>\nA<sub>2<\/sub> = Area of the region representing Red scoring area<br \/>\n= \u03c0(2r)<sup>2<\/sup> \u2013 \u03c0r<sup>2<\/sup> = 3\u03c0r<sup>2<\/sup> = 3A<sub>1<\/sub> = 3 \u00d7 346.5 cm<sup>2<\/sup><br \/>\n= 1039.5 cm<sup>2<\/sup><br \/>\nA<sub>3<\/sub> = Area of the region representing Blue scoring area<br \/>\n= \u03c0 (3r)<sup>2<\/sup> \u2013 \u03c0(2r)<sup>2<\/sup> = 9\u03c0r<sup>2<\/sup> \u2013 4\u03c0r<sup>2<\/sup> = 5\u03c0r<sup>2<\/sup> = 5A<sub>1<\/sub><br \/>\n= 5 \u00d7 346.5 cm<sup>2<\/sup> = 1732.5 cm<sup>2<\/sup><br \/>\nA<sub>4<\/sub> = Area of the region representing Black scoring area<br \/>\n= \u03c0 (4r)<sup>2<\/sup> \u2013 \u03c0(3r)<sup>2<\/sup> = 7\u03c0r<sup>2<\/sup><br \/>\n= 7 A<sub>1<\/sub> = 7 \u00d7 346.5 cm<sup>2<\/sup> = 2425.5 cm<sup>2<\/sup><br \/>\nA<sub>5<\/sub> = Area of the region representing White scoring area<br \/>\n= \u03c0(5r)<sup>2<\/sup> \u2013 \u03c0(4r)<sup>2<\/sup> = 9\u03c0r<sup>2<\/sup><br \/>\n= 9 A<sub>1<\/sub> = 9 \u00d7 346.5 cm<sup>2<\/sup> = 3118.5 cm<sup>2<\/sup><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How To Calculate The Area Of A Circle A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point always remains same. The fixed point is called the centre and the given constant distance is known as the radius of the circle. &#8230; <a title=\"How To Calculate The Area Of A Circle\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/area-of-a-circle\/\" aria-label=\"More on How To Calculate The Area Of A Circle\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[283,279,282,278,280],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>How To Calculate The Area Of A Circle - CBSE Library<\/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:\/\/cbselibrary.com\/area-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 Calculate The Area Of A Circle\" \/>\n<meta property=\"og:description\" content=\"How To Calculate The Area Of A Circle A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point always remains same. The fixed point is called the centre and the given constant distance is known as the radius of the circle. ... 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