Question

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?

Answer 1

see explanation

Explanation:

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%`