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 $\sqrt3= 1.73205$).
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AcademicMathematicsNCERTClass 10

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 $1620.5\ cm^2$

raja
Updated on 10-Oct-2022 13:24:14

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