{"id":639333,"date":"2023-06-26T06:05:54","date_gmt":"2023-06-26T00:35:54","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/in-the-figure-find-the-area-of-the-shaded-region-enclosed-between-two-concentric-circles-of-radii-7-cm-and-14-cm-where-%e2%88%a0aoc\/"},"modified":"2025-06-26T14:43:02","modified_gmt":"2025-06-26T09:13:02","slug":"in-the-figure-find-the-area-of-the-shaded-region-enclosed-between-two-concentric-circles-of-radii-7-cm-and-14-cm-where-%e2%88%a0aoc","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/in-the-figure-find-the-area-of-the-shaded-region-enclosed\/","title":{"rendered":"In the figure, find the area of the shaded region. enclosed between two concentric circles of radii 7\u00a0cm \u00a0 and 14\u00a0cm \u00a0 where \u2220AOC= 40 \u00b0 . \u00a0"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"In the figure, find the area of the shaded region. enclosed between two concentric circles of radii 7\u00a0cm \u00a0 and 14\u00a0cm \u00a0 where \u2220AOC= 40 \u00b0 . \u00a0","custom_permalink":"question\/mathematics\/in-the-figure-find-the-area-of-the-shaded-region-enclosed\/"},"categories":[],"acf":{"question":"

In the figure, find the area of the shaded region. enclosed between two concentric circles of radii <\/span>\r\n\r\n \r\n \r\n 7<\/m:mn> <\/m:mo>cm<\/m:mtext><\/m:mrow>\r\n <\/m:semantics><\/math>  and <\/span>\r\n\r\n \r\n \r\n 14<\/m:mn> <\/m:mo>cm<\/m:mtext><\/m:mrow>\r\n <\/m:semantics><\/math>  where <\/span>\r\n\r\n \r\n \r\n \u2220<\/m:mo>AOC<\/m:mtext>=<\/m:mo>\r\n \r\n 40<\/m:mn><\/m:mrow>\r\n \u00b0<\/m:mo>\r\n <\/m:msup>\r\n .<\/m:mo><\/m:mrow>\r\n <\/m:semantics><\/math> <\/p>

\"In<\/p>

<\/p>

<\/p><\/spanstyle=\"-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal\"><\/spanstyle=\"-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal\">","options":[{"option":" 500.21<\/m:mn>c<\/m:mi> m<\/m:mi> 2<\/m:mn> <\/m:msup> <\/m:mrow> <\/m:semantics><\/math> ","correct":false},{"option":" 440.99<\/m:mn>c<\/m:mi> m<\/m:mi> 2<\/m:mn> <\/m:msup> <\/m:mrow> <\/m:semantics><\/math> ","correct":false},{"option":" 521.77<\/m:mn>c<\/m:mi> m<\/m:mi> 2<\/m:mn> <\/m:msup> <\/m:mrow> <\/m:semantics><\/math> ","correct":false},{"option":" 410.67<\/m:mn>c<\/m:mi> m<\/m:mi> 2<\/m:mn> <\/m:msup> <\/m:mrow> <\/m:semantics><\/math>  <\/span>","correct":true}],"solution":"Given that the radius (r) of inner circle <\/span>\r\n\r\n \r\n \r\n =<\/m:mo>7<\/m:mn>cm<\/m:mi>.<\/m:mo><\/m:mrow>\r\n <\/m:semantics><\/math>  <\/span>Given that the radius (R) of the outer circle <\/span>\r\n\r\n \r\n \r\n =<\/m:mo>14<\/m:mn>cm<\/m:mi>.<\/m:mo><\/m:mrow>\r\n <\/m:semantics><\/math>  <\/span>Areas of sectors OAC and OBD are calculated as: <\/span>
Angle made by the sector is given as<\/span>\r\n\r\n \r\n \r\n =<\/m:mo>\r\n \r\n 40<\/m:mn><\/m:mrow>\r\n \u00b0<\/m:mo>\r\n <\/m:msup>\r\n .<\/m:mo><\/m:mrow>\r\n <\/m:semantics><\/math>  <\/span>\"InWe know that,<\/span>
Area of sector<\/mi>=<\/mo>\u03b8\u03c0<\/mi>r<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac><\/math>, where r is the radius of the circle.<\/span>
Therefore,<\/span>
Area of sector OAC<\/mi>=<\/mo>4<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>\u00d7<\/mo>\u03c0<\/mi>R<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
\u21d2<\/mo>Area of sector OAC<\/mi>=<\/mo>4<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>\u00d7<\/mo>22<\/mn><\/mrow>7<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>(<\/mo>14<\/mn>)<\/mo><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
\u21d2<\/mo>Area of sector OAC<\/mi>=<\/mo>4<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>\u00d7<\/mo>22<\/mn><\/mrow>7<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>196<\/mn><\/math>
\u21d2<\/mo>Area of sector OAC<\/mi>=<\/mo>1<\/mn><\/mrow>9<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>22<\/mn>\u00d7<\/mo>28<\/mn><\/math>
\u21d2<\/mo>Area of sector OAC<\/mi>=<\/mo>68<\/mn>.<\/mo>44<\/mn> c<\/mi>m<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
Area of sector OAC<\/mi>=<\/mo>4<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>\u00d7<\/mo>\u03c0<\/mi>R<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
\u21d2<\/mo>Area of sector OBD<\/mi>=<\/mo>4<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>\u00d7<\/mo>22<\/mn><\/mrow>7<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>(<\/mo>7<\/mn>)<\/mo><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
\u21d2<\/mo>Area of sector OBD<\/mi>=<\/mo>4<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow>36<\/mn>0<\/mn><\/mrow>o<\/mi><\/mrow><\/msup><\/mrow><\/mfrac>\u00d7<\/mo>22<\/mn><\/mrow>7<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>49<\/mn><\/math>
\u21d2<\/mo>Area of sector OBD<\/mi>=<\/mo>1<\/mn><\/mrow>9<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>22<\/mn>\u00d7<\/mo>7<\/mn><\/math>
\u21d2<\/mo>Area of sector OBD<\/mi>=<\/mo>17<\/mn>.<\/mo>11<\/mn> c<\/mi>m<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
From the figure,<\/span>
Area of the small region ABCD <\/mi>=<\/mo> Area of the small sector OCD \u2013 Area of the small sector OBD<\/mi><\/math> <\/span>\u21d2<\/mo>Area of the small region ABCD <\/mi>=<\/mo> <\/mi>68<\/mn>.<\/mo>44<\/mn> \u2013 <\/mi>17<\/mn>.<\/mo>11<\/mn><\/math>
\u21d2<\/mo>Area of the small region ABCD <\/mi>=<\/mo>51<\/mn>.<\/mo>33<\/mn> c<\/mi>m<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
From the figure,<\/span>
Area of shaded region ABDC<\/mi>=<\/mo> Area of the outer circle<\/mi>-<\/mo>Area of inner circle<\/mi>-<\/mo>area of ABCD<\/mi><\/math> <\/span>We know that,<\/span>
Area of circle<\/mi>=<\/mo>\u03c0<\/mi>r<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>, where r is the circle.<\/span>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> \u03c0<\/mi>R<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>\u03c0<\/mi>r<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>51<\/mn>.<\/mo>33<\/mn><\/math>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> \u03c0<\/mi>(<\/mo>14<\/mn>)<\/mo><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>\u03c0<\/mi>(<\/mo>7<\/mn>)<\/mo><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>51<\/mn>.<\/mo>33<\/mn><\/math>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> \u03c0<\/mi>\u00d7<\/mo>196<\/mn>-<\/mo>\u03c0<\/mi>\u00d7<\/mo>49<\/mn>-<\/mo>51<\/mn>.<\/mo>33<\/mn><\/math>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> <\/mi>22<\/mn><\/mrow>7<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>196<\/mn>-<\/mo>22<\/mn><\/mrow>7<\/mn><\/mrow><\/mfrac>\u00d7<\/mo>49<\/mn>-<\/mo>51<\/mn>.<\/mo>33<\/mn><\/math>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> <\/mi>22<\/mn>\u00d7<\/mo>28<\/mn>-<\/mo>22<\/mn>\u00d7<\/mo>7<\/mn>-<\/mo>51<\/mn>.<\/mo>33<\/mn><\/math>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> <\/mi>616<\/mn>-<\/mo>154<\/mn>-<\/mo>51<\/mn>.<\/mo>33<\/mn><\/math>
\u21d2<\/mo>Area of shaded region ABDC<\/mi>=<\/mo> <\/mi>410<\/mn>.<\/mo>67<\/mn> c<\/mi>m<\/mi><\/mrow>2<\/mn><\/mrow><\/msup><\/math>
The required area is equal to <\/span>\r\n\r\n \r\n \r\n 410.67<\/m:mn>c<\/m:mi>\r\n m<\/m:mi>\r\n 2<\/m:mn>\r\n <\/m:msup>\r\n .<\/m:mo><\/m:mrow>\r\n <\/m:semantics><\/math>  <\/span>Hence, option 4 is the correct answer.<\/span>
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