Hexagon

The largest hexagon leaves $3$ small equilateral triangles outside of it and is made up of $6$ small triangles of the same size.

Area of one of the triangles is $\dfrac{320}{6}$

Area of the original triangle is $9$ small triangles, or $9\times\dfrac{320}{6}=\boxed{480}$

Area of regular hexagon = 2.598 * side^2

320 = 2.598 * a^2

a = 11.1

To maximize area of hexagon inside equilateral triangle 3 sides must coincide with the sides of the triangle.

Three equilateral triangles are formed that are connected to the hexagon since the angles adjacent to the hexagon are 60 degrees.

Area of equilateral triangle = 0.433 * side^2

A = 0.433 * 11.1^2 = 53.35

3*A = 160.05

160 + 320 = 480