Formula for the height of an equilateral triangle

In the diagram below, we have six mutually tangent unit circles. What is the area of the shaded region?

Method 1:

Draw \(ABCD\) and altitude \(\overline{BD}.\) Since \(AB = 4,\) and \(AD = 2,\) \(BD = 2\sqrt{3}.\) The area of the shaded region is equal to the area of the parallelogram, minus the areas of the six circle parts inside the parallelogram. The top circle parts are 120^\circ and 60^\circ sectors, from left to right. Together that's a sector of \(180^\circ.\) We have the same situation on the bottom of the parallelogram, and so that's one full circle. Notice the middle two sectors are each \(180^\circ,\) and so that makes another full circle. Thus, we subtract the area of two unit circles from the area of the parallelogram, to obtain \(4\sqrt{3} - 2\pi.\)

Method 2:

Move the top right circle to the lower right. This merely moves the top right shaded region to the lower right. Connect the circle centers to form an equilateral triangle, then subtract the areas of the sectors to find the area shaded.

Method 3:

Join the centers of 3 mutually tangent circles. One shaded region is the area of an equilateral triangle, minus three \(60^\circ\) sectors. Now multiply by 4, because there are 4 such shaded regions.