Since the radius has length 4, and the triangle has sides of length 4, we can see we have the following equilateral triangle in our diagram:



Since we have an equilateral triangle, each angle inside the triangle must be 60° as follows:



Now we'll find the area of the green sector below:

Area of sector \(=(\frac{60}{360})(\pi)(4^2)\)
\(=(\frac{1}{6})(\pi)(16)\)


Now we'll find the area of the red equilateral triangle below

We'll use the formula for the area of an equilateral triangle: Area \(= (\frac{\sqrt{3}}{4})(side^2)\)
\(= (\frac{\sqrt{3}}{4})(4^2)\)
\(= (\frac{\sqrt{3}}{4})(16)\)
\(= 4\sqrt{3}\)

The area of the shaded region = (area of green sector) - (area of red equilateral triangle)
\(=(\frac{1}{6})(\pi)(16)-4\sqrt{3}\)
\(≈1.45\)

A is closest.

Answer: A