Question

Find the area of the shaded region in fig, where circular arc of radius is \[6{\text{ cm}}\] has been drawn with vertex \[O\] of an equilateral triangle \[OAB\] of side \[12\,{\text{cm}}\] as centre.
               

Answer 1

Hint: Before finding the area of the shaded region we have to find the area of the major arc and the area of the equilateral triangle.
We know that, if \[r\] be the circular arc of radius, the area of the major arc is \[\dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\]
We know that the area of an equilateral triangle with each side \[a\] is \[\dfrac{{\sqrt 3 }}{4}{a^2}\] sq. cm.
\[{\text{The area of the shaded part = The area of the equilateral triangle + The area of the major sector}}\]

Complete step-by-step answer:
It is given that; the length of the circular arc of radius is \[6\] cm. Each side of the equilateral triangle is \[12\] cm.
We have to find the area of the shaded region.
From the figure we get,
\[{\text{The area of the shaded part = The area of the equilateral triangle + The area of the major sector}}\]
To find the area of the shaded region we need to find the area of the equilateral triangle and the area of the major sector.
We know that, if \[r\] be the circular arc of radius, the area of the major arc is \[\dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\]
Here, the length of the circular arc of radius is \[6\] cm.
So, we have, \[r = 6,\theta = {300^ \circ }\]
Substitute these values in the general formula of the major arc we get,
\[\dfrac{{{{300}^ \circ }}}{{{{360}^ \circ }}} \times \pi \times {6^2}\] sq. cm
Simplifying we get,
The area of the major arc is \[94.2\] sq. cm
We know that the area of an equilateral triangle with each side \[a\] is \[\dfrac{{\sqrt 3 }}{4}{a^2}\] sq. cm.
Substitute \[a = 12\] we get,
The area of the equilateral triangle is \[\dfrac{{\sqrt 3 }}{4} \times {12^2}\] sq. cm.
Simplifying we get,
The area of the equilateral triangle is \[62.352\] sq. cm.
The area of the shaded region is \[62.352 + 94.2\] sq. cm
Simplifying we get,
The area of the shaded region is \[156.55\] sq. cm
Hence, the area of the shaded region is \[156.55\] sq. cm.

Note: We have taken \[\pi = 3.14\] \[\&\] \[\sqrt 3 = 1.73\]
If \[r\] be the circular arc of radius, the area of the major arc is \[\dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\]
The area of an equilateral triangle with each side \[a\]is \[\dfrac{{\sqrt 3 }}{4}{a^2}\] sq. cm.