Answer key: Volume & Surface Area Worksheet

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Goal: Find the volume and surface area of prisms, cylinders, cones and spheres, and discover how the shapes and their dimensions are related.

Connects to Common Core 7.G.B.6 (Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects.); 8.G.C.9 (Know the formulas for the volumes of cones, cylinders and spheres and use them to solve problems.)

Completed results table

Values are the readouts of the tool (Volume calculator) for each setting.

Answers

1. Find the volume and surface area of the 6 × 4 × 3 prism by hand.

   V = 6 × 4 × 3 = 72 cm³. S = 2(6 × 4 + 6 × 3 + 4 × 3) = 2(24 + 18 + 12) = 108 cm².

2. The cone and the first cylinder have the same radius and height. How do their volumes compare?

   The cone holds 75.398 cm³, exactly one third of the cylinder’s 226.195 cm³, because V = ⅓πr²h for a cone and πr²h for a cylinder.

3. When the cylinder’s radius doubled from 3 to 6 (same height), by what factor did the volume grow? Why?

   From 226.195 to 904.779 cm³: 4 times as much. The volume formula uses r², and 2² = 4.

4. The sphere with radius 5 fits exactly inside the cylinder with radius 5 and height 10. What fraction of the cylinder does the sphere fill?

   523.599 ÷ 785.398 ≈ 0.667, so the sphere fills two thirds of the cylinder, a result Archimedes was proud of.

5. Find the sphere’s volume by hand with V = ⁴⁄₃πr³.

   ⁴⁄₃ × π × 5³ = ⁴⁄₃ × π × 125 ≈ 523.599 cm³.

6. A juice company can sell its juice in a cylinder (radius 3 cm, height 8 cm) or a box (6 cm × 6 cm × 8 cm). Which holds more, and which needs more packaging material?

   The box holds more (288 cm³ versus 226.195 cm³), but it also needs more material (264 cm² versus 207.345 cm²).