Question

Three circles each of radius r units are drawn inside an
equilateral triangle of side 'a' units, such that each circle
touches the other two and two sides of the triangle as shown
in the figure. (P, Q and R are the centres of the three circles).
Then relation between r and a is

Answer 1

The image given below is a representation of the question.

In the left bottom circle, draw a line from the centre of the circle perpendicular to the side of the angle. This distance will be r, the radius of the circle. let x be the distance between the vertex of the triangle and the line just drawn. Let's call this triangle AOB, where A is the vertex of the triangle, O is the centre of the circle and B is the point where our new line meets the side of the triangle.
As the triangle is an equilateral triangle, each angle should be 60 degrees.
Thus, Angle OAB = 30 degrees

Thus, Tan 30 = r/x
Tan 30. x = r
x = r/ Tan 30

We know that
a = 2x + 2r
   = 2r/ Tan 30 +2r
   = 2r ( 1/ Tan 30 + 1)
   = 2r ( 1/(1/3^1/2) + 1)
   = 2r ( 3^1/2 + 1)
a/r = 2(3^1/2 + 1)

This is the relation between r and a.