If we connect the centers of the three smaller circles, we get an equilateral triangle because the length of each side is twice the radius of a smaller circle. Then if we connect the center of the equilateral triangle to each of its three vertices, we get three new congruent triangles. Since these new triangles are congruent to each other,
m∠CBD = (1/2)(60°) = 30°
And let's say
radius of smaller circle = AB = BD = r
And so △CBD is a 30-60-90 triangle where
the side across from the 60° angle = r
the side across from the 30° angle = r / √3
the side across from the 90° angle = 2r / √3
BC = 2r / √3
Because the radius of the bigger circle = 1
AB + BC = 1
r + 2r / √3 = 1
r( 1 + 2/√3 ) = 1
r = 1 / ( 1 + 2/√3 )
r = 2√3 - 3
