the area of an equilateral triangle is 49 root 3 cm^{2}taking each angular point as centre,circles are drawn with radii equal to half of the legth of the side of the triangle.find the area of the triangle not included in the circles. (take pi root3=1.73)

Usha Kalva Kalva , Meritnation Expert added an answer, on 31/12/13

Let "a" be the side of equilateral triangle.

$\frac{\sqrt{3}{a}^{2}}{4}=49\sqrt{3};\phantom{\rule{0ex}{0ex}}{a}^{2}=49*4;\phantom{\rule{0ex}{0ex}}a=7*2=14cm;$

Radius of circle = 14/2 = 7 cm

Area of the first circle occupied by triangle = area of sector

$=\frac{1}{2}{r}^{2}\theta =\frac{1}{2}*{7}^{2}*\pi *\frac{60}{180}=\frac{77}{3}c{m}^{2}$

Area of all the 3 sectors = 77/3 * 3 = 77 cm^{2}

Area of triangle not included in the circle

= area of triangle- area of all the 3 sectors

=$=49\sqrt{3}-77=7.87c{m}^{2}$

$\frac{\sqrt{3}{a}^{2}}{4}=49\sqrt{3};\phantom{\rule{0ex}{0ex}}{a}^{2}=49*4;\phantom{\rule{0ex}{0ex}}a=7*2=14cm;$

Radius of circle = 14/2 = 7 cm

Area of the first circle occupied by triangle = area of sector

$=\frac{1}{2}{r}^{2}\theta =\frac{1}{2}*{7}^{2}*\pi *\frac{60}{180}=\frac{77}{3}c{m}^{2}$

Area of all the 3 sectors = 77/3 * 3 = 77 cm

Area of triangle not included in the circle

= area of triangle- area of all the 3 sectors

=$=49\sqrt{3}-77=7.87c{m}^{2}$

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