In the figure below, $ A B C $ is an equilateral triangle of side $ 8 \mathrm{~cm} . A, B $ and $ C $ are the centres of circular arcs of radius $ 4 \mathrm{~cm} $. Find the area of the shaded region correct upto 2 decimal places. (Take $ \pi=3.142 $ and $ \sqrt{3}=1.732 $ ). "
Given:
\( A B C \) is an equilateral triangle of side \( 8 \mathrm{~cm} . A, B \) and \( C \) are the centres of circular arcs of radius \( 4 \mathrm{~cm} \).
To do:
We have to find the area of the shaded region correct upto 2 decimal places.
Solution:
Length of each side of $\triangle ABC = 8\ cm$
Area of the triangle $=\frac{\sqrt{3}}{4} a^{2}$
$=\frac{\sqrt{3}}{4}(8)^{2}$
$=\frac{1.732 \times 64}{4}$
$=1.732 \times 16$
$=27.712 \mathrm{~cm}^{2}$
Angle of each sector $=60^{\circ}$
Area of three sectors $=3 \times \pi r^{2} \times \frac{60^{\circ}}{360^{\circ}}$
$=3 \times 3.142 \times 4 \times 4 \times \frac{1}{6}$
$=1.571 \times 16$
$=25.136 \mathrm{~cm}^{2}$
Therefore,
Area of the shaded region $=$ Area of $\Delta \mathrm{ABC}-$ Areas of three sectors
$=27.712-25.136$
$=2.576 \mathrm{~cm}^{2}$
The area of the shaded region correct upto 2 decimal places is $2.58 \mathrm{~cm}^{2}$.
Related Articles In the below figure, an equilateral triangle \( A B C \) of side \( 6 \mathrm{~cm} \) has been inscribed in a circle. Find the area of the shaded region. (Take \( \pi=3.14) \)"\n
Find the area of a shaded region in the below figure, where a circular arc of radius \( 7 \mathrm{~cm} \) has been drawn with vertex \( A \) of an equilateral triangle \( A B C \) of side \( 14 \mathrm{~cm} \) as centre. (Use \( \pi=22 / 7 \) and \( \sqrt{3}=1.73) \)"\n
In the figure below, two circles with centres \( A \) and \( B \) touch each other at the point \( C \). If \( A C=8 \mathrm{~cm} \) and \( A B=3 \mathrm{~cm} \), find the area of the shaded region."\n
In the figure below, \( A B C D \) is a square with side \( 2 \sqrt{2} \mathrm{~cm} \) and inscribed in a circle. Find the area of the shaded region. (Use \( \pi=3.14) \)"\n
Find the area of the shaded region in the below figure, if \( A C=24 \mathrm{~cm}, B C=10 \mathrm{~cm} \) and \( O \) is the centre of the circle. (Use \( \pi=3.14) \)"\n
In the below figure, \( A B C D \) is a trapezium of area \( 24.5 \mathrm{~cm}^{2} . \) In it, \( A D \| B C, \angle D A B=90^{\circ} \), \( A D=10 \mathrm{~cm} \) and \( B C=4 \mathrm{~cm} \). If \( A B E \) is a quadrant of a circle, find the area of the shaded region. (Take \( \pi=22 / 7) \)."\n
In the below figure, \( O A C B \) is a quadrant of a circle with centre \( O \) and radius \( 3.5 \mathrm{~cm} \). If \( O D=2 \mathrm{~cm} \), find the area of the shaded region."\n
In the figure, a \( \triangle A B C \) is drawn to circumscribe a circle of radius \( 4 \mathrm{~cm} \) such that the segments \( B D \) and \( D C \) are of lengths \( 8 \mathrm{~cm} \) and \( 6 \mathrm{~cm} \) respectively. Find the lengths of sides \( A B \) and \( A C \), when area of \( \triangle A B C \) is \( 84 \mathrm{~cm}^{2} \). "\n
Find the perimeter of each of the following shapes :(a) A triangle of sides \( 3 \mathrm{~cm}, 4 \mathrm{~cm} \) and \( 5 \mathrm{~cm} \).(b) An equilateral triangle of side \( 9 \mathrm{~cm} \).(c) An isosceles triangle with equal sides \( 8 \mathrm{~cm} \) each and third side \( 6 \mathrm{~cm} \).
In the figure below, \( A B C D \) is a trapezium with \( A B \| D C, A B=18 \mathrm{~cm}, D C=32 \mathrm{~cm} \) and the distance between \( A B \) and \( D C \) is \( 14 \mathrm{~cm} \). Circles of equal radii \( 7 \mathrm{~cm} \) with centres \( A, B, C \) and \( D \) have been drawn. Then, find the area of the shaded region of the figure. (Use \( \pi=22 / 7) \)."\n
In the below figure, \( A B C \) is a right-angled triangle, \( \angle B=90^{\circ}, A B=28 \mathrm{~cm} \) and \( B C=21 \mathrm{~cm} \). With \( A C \) as diameter a semicircle is drawn and with \( B C \) as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places."\n
In the below figure, \( O \) is the centre of a circular arc and \( A O B \) is a straight line. Find the perimeter and the area of the shaded region correct to one decimal place. (Take \( \pi=3.142) \)"\n
A circle is inscribed in an equilateral triangle \( A B C \) is side \( 12 \mathrm{~cm} \), touching its sides. Find the radius of the inscribed circle and the area of the shaded part."\n
In Fig 4, a circle is inscribed in an equilateral triangle $\vartriangle ABC$ of side $12\ cm$. Find the radius of inscribed circle and the area of the shaded region. [$Use\ \pi =3.14\ and\ \sqrt{3} =1.73$]."\n
In the below figure, \( A B C \) is a right angled triangle in which \( \angle A=90^{\circ}, A B=21 \mathrm{~cm} \) and \( A C=28 \mathrm{~cm} . \) Semi-circles are described on \( A B, B C \) and \( A C \) as diameters. Find the area of the shaded region."\n
Kickstart Your Career
Get certified by completing the course
Get Started