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

For side length s the area of the triangle is
.. A = (√3)/4*s^2
The radius of the circle will be
.. r = s/(2√3)
so the area of the circle is
.. A = π*r^2 = π*s^2/12

The fraction of the triangle that is not in the circle is
.. 1 - (π/12)/((√3)/4)
.. = 1 -π/(3√3)
.. ≈ 39.54%