If the apothem is 6.9 mm and a side length is 8 mm what is the area of the hexagon

Hex = 6
draw it and draw lines from each corner to every other corner... you're left with 6 triangles
work out the area of one of these triangles from your question... area of hexagon = 6 x area of a triangle

If you join the 3 diagonals of a regular hexagone, you'll divide it into 6 equilateral triangles. Let's calculate the area of one equilateral triangle:

side = 8 mm, 
apothem (= Altitude) = 6.9 mm, 
since the area of a triangle = side x altitude (opposed to this side)/2, we can write:

Area = (8 x 6.9)/2, but we have 6 identical triangles, then Total area :
Total Area = 6[(8 x 6.9)/2]/2 = 166 mm² (rounded to the nearest unit)