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The area of an equilateral triangle ABC is $17320.5\ cm^2$. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see figure). Find the area of the shaded region.
(Use $\pi = 3.14$ and $\sqrt3= 1.73205$).
"
Given:
The area of an equilateral triangle ABC is $17320.5\ cm^2$. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle.
To do:
We have to find the area of the shaded region.
Solution:
Area of the equilateral triangle $ABC = 17320.5\ cm^2$
Let the side of the triangle $ABC$ be $a$
This implies,
Area of $\triangle ABC = \frac{\sqrt3}{4}a^2$
$\frac{\sqrt3}{4}a^2= 17320.5$
$ a^{2} =\frac{17320.5 \times 4}{\sqrt{3}}$
$a^{2}=\frac{17320.5 \times 4}{1.73205}$
$a^{2}=40000$
$a=200 \mathrm{~cm}$
Radius of the circle drawn at each vertex $=\frac{1}{2} \text { (side of the equilateral triangle) }$
$=\frac{1}{2} \times 200$
$=100 \mathrm{~cm}$
Area of the sector formed at each vertex having radius $100 \mathrm{~cm}$ and sector angle $60^{\circ}=3.14 \times \frac{100 \times 100 \times 60^{\circ}}{360^{\circ}}$
$=\frac{3.14 \times 100 \times 100}{6}$
$=\frac{31400}{6}$
Area of all 3 sectors $=\frac{3 \times 31400}{6}$
$=15700 \mathrm{~cm}^{2}$
Therefore,
Area of the shaded portion $=$ Area of the equilateral triangle
$-$ Area of the three sectors formed at each vertex
$= 17320.5 - 15700$
$= 1620.5\ cm^2$
The area of the shaded region is $1620.5\ cm^2$