PS - Geometry

i have no clue on this question, can someone please explain why 6 rectangles? 2 Hex??

i was going with the approach that the ball is a sphere and as such its surface area is 4pir^2

so surface area of ball=16pi

Hope visualization might help, hoever I could not get a picture for solid hexagon better than this one.

Draw a regular hexagon and a circle inside it such that the circle touches the opposite sides (since the question is asking for the minimum surface area of the box, diameter of the circular ball should be the same as the distance between two opposite sides of the hexagon.

Now, half of diameter is 2 and since the center of the circle and two adjacant edges of hexagon form an equilaterl triangle, the side of the square = 4/sqrt(3).
Also, since we are talking of the minimum surface area, height of the box should be the same as the diameter of the ball.

Hence, total surface area (of course outside surface area) = 6 sides * 4/sqrt3 * 4 (height of box) = 32 * sqrt(3).
additionally, top and bottom surfaces of the hexagon will have area = 2 (top and bottom) * 6 (equlateral triangles in a hexagon) * 1/2 * 2 * 4/sqrt(3) = 16 * sqrt(3).

Hence, total area = 48 * sqrt(3).