First, let's connect the points of tangency to each other to make a triangle.
Then.....the area in question = the area of this triangle - the areas of the segments
To find the area of each segment, let's draw a circle that is a part of one of these arcs.
Then draw two radii to the points of tangency.
Since the central angle = 60° , and the two blue sides = 4 ,
this is an equilateral triangle where each angle is 60° .
So, the purple sides also are 4 units long.
area of segment = area of sector - area of triangle
area of segment = (60º)(π 42) / 360º - (1/2)(4)(4 sin 60° )
area of segment = 8π / 3 - 4√3
Now, we just found the area of the triangle inside the circle = 4√3 . The area of the triangle outside the circle is also an equilateral triangle with sides 4 units long, so its area also = 4√3 .
the area in question = 4√3 - 3(8π / 3 - 4√3)
= 4√3 - 8π + 12√3
= 16√3 + -8π
And..... 16 + 3 + -8 = 11