The shaded area is then the area of this triangle minus the areas of the three circular sectors inside that triangle.
Herons formula is not required. It provides a method to find the area of a triangle from the three sides but we are dealing here with 30-60-90 triangles where the area is readily calculated using the 2-1- rad3 relationship. prnceps solution has a typo error.Its 60/360 not 360/60 in the area for the three sectors.Google Herons formula to see it
Heron's Formula for area of triangle given three sides
A=sq rt of s(s-a)(s-b)(s-c) where s= 1/2 of perimeter and in my opinion is the long way to find the area of an equilateral triangle. Princips uses the property of a 30-60-90 triiangle whch ought to be known by a geometry student but does not describe it that way