Question

Six circles of radius 1 unit are drawn tangent to the sides of a regular hexagon. Each circle is also tangent to two other circles as shown in the drawing. What is the perimeter of the hexagon? Express your answer in simplest radical form.

Answer 1

Step-by-step explanation:

if we connect center of circles which tangent each other, then we get bigger regular hexagon with side length 2 unit but it differ 60° from original (smaller) hexagon. hence smaller hexagon has height 2•2–2•1 = 2. from center of smaller hexagon draw line to any two neigbouring vertexes so we get an equilateral triangle with height 1. by using pythagorean or trigonometry we get side length of smaller hexagon

$\frac{2}{3} \sqrt{3}$

so the perimeter is

$4 \sqrt{3}$

unit