Question

Find the area of the segment AYB, shown in figure, if radius of the circle is 12 cm and \[\angle AOB={{120}^{\circ }}\]. Use \[\pi =\dfrac{22}{7}\].

Answer 1

Hint: First of all we will find the area of sector AOB and then we subtract the area of triangle AOB to get the required area of segment AYB. The area of sector subtending an angle ‘\[\theta \]’ and radius ‘r’ is given as:
Area \[=\pi {{r}^{2}}\times \left( \dfrac{{{\theta }^{\circ }}}{{{360}^{\circ }}} \right)\]
Also, we know that the area of a triangle having sides a, b and \[\theta \] as the angle between them is equal to \[\left( \dfrac{1}{2}ab\times \sin \theta \right)\].

Complete step-by-step answer:

We have been asked to find the area of segment AYB from the figure given with radius of circle as 21 cm and \[\angle AOB={{120}^{\circ }}\].

First of all we will find the area of sector using the formula \[\pi {{r}^{2}}\times \left( \dfrac{{{\theta }^{\circ }}}{{{360}^{\circ }}} \right)\] where r is the radius of the circle and \[\theta \] is the angle subtended by the arc corresponding to the sector.
Area of sector AOB \[=\pi \times {{\left( 21 \right)}^{2}}\times \dfrac{{{120}^{\circ }}}{{{360}^{\circ }}}\]
Since we have \[\pi =\dfrac{22}{7}\]
\[\Rightarrow \dfrac{22}{7}\times 21\times 21\times \dfrac{1}{3}=462c{{m}^{2}}\]
Now we know the area of a triangle having sides a, b and angle between them as \[\theta \] is given by \[\left( \dfrac{1}{2}ab\times \sin \theta \right)\].
\[\Rightarrow \] Area of \[\Delta ABC\]\[=\dfrac{1}{2}\times 21\times 21\times \sin {{120}^{\circ }}\]
Since \[\sin {{120}^{\circ }}=\dfrac{\sqrt{3}}{2}\] and \[\sqrt{3}=1.732\]
\[\Rightarrow \dfrac{1}{2}\times 21\times 21\times \dfrac{\sqrt{3}}{2}=\dfrac{441\times 1.732}{4}=190.95c{{m}^{2}}\] (approx.)
For the figure we can say that the area of segment AYB is equal to the difference of area of sector AOB and area of \[\Delta ABC\].
\[\Rightarrow \]Area of segment AYB \[=\left( 462-190.95 \right)c{{m}^{2}}=271.05c{{m}^{2}}\]
Therefore, the area of the segment is equal to \[271.05c{{m}^{2}}\] approximately.

Note: Be careful while doing calculation and substitute the value of pi \[\left( \pi \right)\] is equal to \[\dfrac{22}{7}\] which is given to us. Don’t use the value of pi \[\left( \pi \right)\] as 3.14 instead of \[\dfrac{22}{7}\]. Also, remember that, the area of a triangle when we have been given 2 sides a, b and the angle between them as ‘\[\theta \]’ is given by \[\dfrac{1}{2}\times \left( ab \right)\times \sin \theta \].