Question

Find the area of the segment $ AYB $ show in the adjacent fig. Give radius of the circle is \[21cm\] and \[\angle AOB = 120^\circ \left( {use\,\pi = \dfrac{{22}}{7},\sqrt 3 = 1.732} \right)\]

Answer 1

Hint: To find area of segment AYB, we use area segment formula of mensuration. As Y lies on a minor segment of chord AB. Therefore, we calculate minor segment area formed by chord AB and use an angle of $ {120^0} $ to get the required area of segment.
Area of segment = $ \dfrac{\theta }{{{{360}^0}}} \times \pi {r^2} - \dfrac{1}{2}{r^2}.\sin \theta $ , where $ \theta $ an angle at center and r is radius of the circle.

Complete step-by-step answer:
Here, from figure we see that angle formed by chord AB at centre of the circle is $ {120^0} $
And the radius of the circle is $ 21cm $ .

Area of minor segment is given as: $ \dfrac{\theta }{{{{360}^0}}} \times \pi {r^2} - \dfrac{1}{2}{r^2}.\sin \theta $
Substituting values of $ \theta \,\,and\,\,radius $ in above formula we have:
Area = $ \dfrac{{{{120}^0}}}{{{{360}^0}}} \times \dfrac{{22}}{7} \times {(21)^2} - \dfrac{1}{2} \times {(21)^2} \times \sin ({120^0}) $
On simplifying the right hand side of the above equation. We have,
 $ \Rightarrow Area = \dfrac{1}{3} \times \dfrac{{22}}{7} \times 21 \times 21 - \dfrac{1}{2} \times 441 \times \left( {\dfrac{{\sqrt 3 }}{2}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\sin {{120}^0} = \dfrac{{\sqrt 3 }}{2}} \right\}\,\,\,\, $
 $ Area = 22 \times 21 - \dfrac{1}{4} \times 441 \times (1.732),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\sqrt 3 = 1.732} \right\} $
 $
\Rightarrow Area = 462 - 110.25 \times 1.732 \\
\Rightarrow Area = 462 - 190.95 \\
\Rightarrow Area = 271.05 \;
  $
Therefore, the area of the minor segment of chord AB is $ 271.05\,c{m^2} $ .

Note: Chord of a circle divides the circle into two segment areas one is major segment area and other is minor segment area. Also, area of segment can be calculated by calculating the difference of area of sector formed by chord and area of triangle. There are different ways to calculate the area of a triangle depending upon the type of triangle given in the problem. Also, we can find the area of segment by using the direct formula of mensuration mentioned above.