The figure below shows 13 circles of radius 1 within a larger circle.

The four circles can be viewed as the sides of an equilateral as shown in the figure below.

The center of the bigger circle, is the centroid of the equilateral triangle.

Hence, the length of AO is \(\frac{\sqrt 3 * S}{ 3}\)

S - Side of the equilateral triangle

The side of the equilateral triangle is equal to the sum of diameter of two circles and radius of the other two circles where the vertex of the triangle lies.

Thus S = 2 + 2 + 1 + 1 = 6

AO = \(\frac{\sqrt 3 * 6}{ 3}\)

AO = \( 2 * \sqrt 3 \)

The radius if the bigger circle is OZ

OZ = AO + AZ

OZ = \( 2 * \sqrt 3 \) + 1

Area of the shaded region = Area of the bigger circle - ( 13 * Area of the smaller circles)

Area of the shaded region = \( \pi (2 * \sqrt 3 + 1) ^ 2 \) - (13 * \( \pi (1) ^ 2 \))

Area of the shaded region = \( 12\pi + 1\pi + 4\sqrt 3 \pi - 13 \pi \)

Area of the shaded region = \( 4\sqrt 3 \pi \)