L3 · How surface area and volume relate when a shape changes

Learning goal

“I can compare solids that hold the SAME volume and explain their surface areas: joining parts hides surface, splitting a solid creates new surface, changing a shape's proportions changes surface area even when the volume does not move - and among rectangular prisms with the same volume, the cube has the smallest surface area.”

Constant volume — a comparison condition: the solids being compared hold exactly the same volume; only their shape (and so their surface area) varies.

Minimize — to make as small as possible. Packaging designers minimize surface area (material) for a required volume (contents).

Interior face — a face created INSIDE a solid when it is cut, or hidden inside a composite when parts join. Interior faces are not surface - until a cut exposes them.

Worked example

Three rectangular boxes each hold exactly 64 cm3. Box 1: 8 x 8 x 1. Box 2: 8 x 4 x 2. Box 3: 4 x 4 x 4. Step 1: Confirm the volumes: 8 x 8 x 1 = 64; 8 x 4 x 2 = 64; 4 x 4 x 4 = 64 - same contents for all three. Step 2: Surface areas. Box 1: 2(64 + 8 + 8) = 160 cm2. Box 2: 2(32 + 16 + 8) = 112 cm2. Box 3: 2(16 + 16 + 16) = 96 cm2. Step 3: Read the pattern: as the box gets LESS flat and MORE equal-sided, the surface area falls. The cube - all dimensions equal - has the smallest surface area of any rectangular prism with this volume. That is a curriculum fact worth memorizing as a principle: for a fixed volume, the cube minimizes surface area. Step 4: Why it matters: material costs money and conducts heat. A designer who must hold 64 cm3 spends 160 cm2 of material on the flat box and only 96 cm2 on the cube-shaped one.

Check yourself — explain it

Two prisms have the same volume. Prism P is long and thin; prism Q is nearly a cube. Which has the larger surface area?

Paired problem — example still visible

Two boxes hold 27 cm3: box X is 9 x 3 x 1, box Y is 3 x 3 x 3. Find both surface areas and say which is smaller. -> X: 2(27 + 9 + 3) = 78 cm2. Y: 6 x 9 = 54 cm2. The cube Y is smaller.

Faded completion — last step is yours

"Two cubes of edge 3 cm stand apart: each has SA = 54, total 108. They are joined face-to-face into a 6 x 3 x 3 prism: SA = 2(18 + 18 + 9) = ___. Joining changed the surface area by 108 - ___ = ___ cm2 (a decrease)." -> 90; 90; 18

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