You are a packaging engineer designing a new shipping box. Use volume and surface area to spec out the perfect box â then decide whether it costs less to build it bigger or more compact.
A small business needs a custom shipping box. Your job is to design it from scratch: compute the volume so the product fits, work with fractional edge lengths to find the exact number of unit cubes that fill the box, unfold the net to find the surface area, and then figure out the cost of the cardboard material. Fill in every field, hit Calculate or Check, complete the checklist, and write your package spec sheet.
The volume of a rectangular prism is the amount of space inside. Use the formula V = l Ă w Ă h to find how many cubic units fill your box.
Multiply all three dimensions together. For a box that is 8 Ă 5 Ă 3, count how many 1-unit cubes fit along each edge, then multiply: 8 Ă 5 = 40 layers on the bottom, and 40 Ă 3 = 120 total cubic units.
Real packages often have fractional measurements. The formula V = l Ă w Ă h works the same way with decimal (fractional) edge lengths â and we can still count unit cubes by thinking about fractional layers.
Treat the decimals as fractions: 2.5 = 5/2, 1.5 = 3/2, 3 = 3/1. Multiply: (5/2) Ă (3/2) Ă (3/1) = 45/4 = 11.25. Each full unit cube fills 1 cubic unit. A half-unit along one edge gives cubes that are each 1/2 a cubic unit. Count them all and they still total V = l Ă w Ă h.
A rectangular prism has 6 faces that form three pairs: top/bottom, front/back, and two sides. Unfold them into a net and add the areas. Formula: SA = 2(lw + lh + wh).
Step 1 â find each face pair: top/bottom = 2 Ă (l Ă w), front/back = 2 Ă (l Ă h), sides = 2 Ă (w Ă h). Step 2 â add all three results. For an 8 Ă 5 Ă 3 box: 2(40) + 2(24) + 2(15) = 80 + 48 + 30 = 158 square units.
Cardboard costs money per square unit. Multiply the surface area by the cost per square unit to find the total material cost for one box. Use your judgment â is the box worth the cost?
Write a short spec sheet (3â5 sentences) that uses your real numbers from above. Describe your box dimensions, volume, surface area, and material cost, and explain why this design is a good choice.
| Category | 4 â Expert | 3 â Proficient | 2 â Developing |
|---|---|---|---|
| Volume (6.G.2) | V = lĂwĂh correct with unit label; unit-cube reasoning explained | Volume calculation correct | Formula used but computation error |
| Fractional Edge Volume (6.G.2) | Fractional edges computed correctly; fractional unit-cube explanation included | Fractional volume correct | Attempted with minor error on fractional arithmetic |
| Surface Area / Nets (6.G.4) | All three face pairs identified and SA = 2(lw+lh+wh) applied correctly | Surface area correct | One face pair missing or minor error |
| Application / Communication | Cost computed correctly; spec sheet justifies design with all numbers | Cost correct; spec sheet uses most numbers | Attempted; spec sheet unclear or missing numbers |