Question

Three circles with radii 1 is circumscribed tightly in an equilateral triangle. Find the perimeter of the triangle

Answer 1

As an illustration of your problem, place 3 identical coins, e.g. 3 quarters
in such a way that they are tangent to each other.
Sketching a triangle around them results in an equilateral triangle, we want the perimeter of that triangle.

Let's concentrate on one of these circles.
From its centre draw a perpendicular to one of its sides and joint the centre to its nearest vertex of the triangle.
clearly we have a 30-60-90 degree right-angled triangle with the side opposite the 20° angle as 1, so the other sides are √3 and 2
The same thing is true for the other two circles.
So, one side of the circumscribing triangle is
√3 + 2 + √3 or 2 + 2√3

So the perimeter of the triangle is
6 + 6√3 or 6(1 + √3) or appr 16.4