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The area of an equilateral triangle is $49 {\sqrt{3} cm}^{2}$ . Taking each angular point as centre, a circle is described with radius equal to half the length of the side of the triangle as shown in figure. What is the area of the triangle not included in the circle? [Take $\sqrt{3} = 1.73$ ]

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{5.67 cm}^{2}

{6.12 cm}^{2}

{7.77 cm}^{2}

{7.98 cm}^{2}

answer is C.

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Detailed Solution

Given,

Area of equilateral

ΔABC = 49 \sqrt{3} {cm}^{2}

.
Taking each angular point as centre, a circle is described with radius equal to half the length of the side of the triangle as shown in figure.
Area of an equilateral triangle is

\frac{\sqrt{3}}{4} \times sid e^{2}

\frac{\sqrt{3}}{4} \times {AB}^{2} = 49 \sqrt{3} {cm}^{2}

\Rightarrow {AB}^{2} = 196

\Rightarrow AB = \sqrt{196} = 14 cm

Now,

\frac{1}{2} AB = AF = FB = BD = DC = CE = EA

= Radius (r).

∴

AF = FB = BD = DC = CE = EA = 7 cm = r

Area of a sector is

\frac{θ}{360 °} \times π r^{2}

.
Now, area of triangle not included

=

Area of the equilateral triangle - 3(Area of the sector BFD).

\Rightarrow

Area of triangle not included =

49 \sqrt{3} - 3 \times \frac{θ}{360 °} \times π r^{2}

\Rightarrow

Area of triangle not included

=

49 \sqrt{3} - 3 \times \frac{60^{0}}{360^{0}} \times π r^{2}

\Rightarrow

Area of triangle not included

= 49

\times

1.73 - 77

\Rightarrow

Area of triangle not included

= 84.77 - 77

\Rightarrow

Area of triangle not included

= 7.77

{cm}^{2}

Hence, option 3 is correct.

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The area of an equilateral triangle is 493cm2. Taking each angular point as centre, a circle is described with radius equal to half the length of the side of the triangle as shown in figure. What is the area of the triangle not included in the circle? [Take 3=1.73]

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The area of an equilateral triangle is $49 {\sqrt{3} cm}^{2}$ . Taking each angular point as centre, a circle is described with radius equal to half the length of the side of the triangle as shown in figure. What is the area of the triangle not included in the circle? [Take $\sqrt{3} = 1.73$ ]