A mandrel designed for use in the construction of a custom-sized box includes a common platform on which a plurality of peripheral members is independently mounted. Each peripheral member comprises a disc-shaped base rotatably mounted on the common platform, a tongue slidably mounted within a channel formed in the base, and a vertical upright rotatably mounted at one end of the tongue. To construct a box, an elongated continuous strip of material is helically wrapped around the vertical uprights to form a main box body with a plurality of bands. Cutting mechanisms in each upright are actuated to create slots in the main box body that define flaps used to selectively enclose the open top and bottom ends of the box body. By reconfiguring the vertical uprights in the mandrel, a customized box can be constructed that is dimensioned to receive a consumer product with limited wasted space.