Packaging Designer

Pre-Algebra · Axiom Academy

LESSON The Packaging Design Challenge A box can hold the same amount of stuff in many shapes — and the shape decides how much cardboard you spend. 1. Same Box of Air, Different Skin Fix the volume at 1,000 in³ and reshape the box: stretch it tall, squash it flat. The air inside never changes — but watch the cardboard meter . It falls to a low point and then climbs again. There is a best shape hiding in the middle. Volume — how much fits inside (held fixed at 1,000) Surface area — how much cardboard the skin needs 2. Unfold It to See the Cardboard Surface area isn't abstract — it's the flat sheet you'd cut to build the box. Unfold a box and you get six rectangles: a top and bottom, two sides, a front and back. Add their areas and you have exactly the cardboard required. A flat box has the same air inside as a cube, yet its unfolded sheet is far bigger. Six identical 100 in² faces. Total sheet: . A huge 500 in² top and bottom dominate. Total sheet: for the same 1,000 in³. Opposite faces match, so SA = 2(LW + LH + WH) — pairs of top/bottom, front/back, left/right. Flattening trades a small footprint for two large faces, so the sheet balloons. The cube's six tiles tile up to 600 in². The flat box's two giant faces alone are 1,000 in² — before you even add its edges — which is why its sheet nearly doubles. 3. The Cube Sits at the Bottom of the Valley

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