A circle is inscribed in an equilateral triangle with a side length measuring 10 mm. Represent the area of the shaded region as a percentage of the total area of the triangle?

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1 Answer
Mar 28, 2018

see explanation

Explanation:

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Area of an equilateral triangle #A_1=sqrt3/4*a^2#, where #a# is the length of one side of the triangle.
given side length #=10# mm
#=> A_1=sqrt3/4*10^2=25sqrt3 " mm^2#,

Let #r# be the inradius of the inscribed circle,
formula for inradius of the incircle of the triangle :
#r=(2A)/p#, where #A and p# are the area and the perimeter of the triangle, respectively,
#=> p=3xx10=30# mm
#=> r= (2A_1)/p=(2*25sqrt3)/30=(5sqrt3)/3# mm
#=># area of the incircle #A_2=pi*r^2=pi*((5sqrt3)/3)^2#
#=(25pi)/3 " mm"^2#

shaded area #=A_s=A_1-A_2=25sqrt3-(25pi)/3=25(sqrt3-pi/3) " mm"^2#

ratio of #(A_s)/(A_1)=(25(sqrt3-pi/3))/(25sqrt3)#
#=(sqrt3-pi/3)/sqrt3~~0.3954~~39.54%#