now, if you check the picture below, the top-side has the base alone, the hexagon, notice, if you divide a circle the 360° of a circle in 6 even angles, like the hexagon does, you end up with 6 angles of 60° each, and 6 equilateral triangles made by them, now, if you run an angle bisector, like we did on the bottom side of the hexagon, you end up splitting the angle in equal halves.
so.. then we end up with the a 30, 60, 90 angles triangle, and thus we apply the 30-60-90 rule, to get the apothem.
now the area of a regular polygon is just (1/2) (apothem)(perimeter).
now, we know the apothem, the perimeter of the hexagon is just 4+4+4+4+4+4. So, just get those values, plug them in to get the area of the hexagonal base.
if you look at the picture bottom-side, the slant-height of the pyramid, is just the altitude/height of the triangular face. So, you really have 6 triangles whose base is 4 and height is 6, and you know how to get the area of a triangle.
so, just get the area of the base, plus the area of the triangular faces, sum them up, and that's the total area of the hexagonal pyramid.
