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Chapter 14 · Period 1 of 16 · 50 min

Deriving Surface Area and Volume of Cuboids and Cubes

Cuboids and Cubes · Deductive

This unit opens the chapter at the point where students have already met nets of solids in the previous grade, so the net of a cuboid (Fig. 14.2) is a natural entry point rather than a novelty. A common error is omitting the factor 2 when pairing opposite faces in the surface-area formula; note that any face may serve as the base, since the formula is symmetric in l, w, h. For self-study, Example 1, p.125 is worth re-reading slowly for the packaging/heat-transfer discussion.

0-12Introduce the cuboid as the three-dimensional version of a rectangle with dimensions l, w, h. Draw and label the net of a cuboid on the board and guide students to add the six face areas, deriving TSA = 2(wl + hl + hw). Establish the cube as the special case giving TSA = 6a² and volume = a³.
12-25Define volume using the unit-cube model and develop V = lwh by picturing the cuboid as thin rectangular sheets stacked like a pack of playing cards. Walk through Example 1, p.125: compare a 6 cm cube (SA = 216 cm², V = 216 cm³) with a 9 cm × 6 cm × 4 cm cuboid (SA = 228 cm², V = 216 cm³), drawing out the engineering implication that for equal capacity a cube uses less surface material.
25-38Students attempt Exercise Set 14.1 Q1, p.125 (cube from volume) and Q6, p.126 (cuboid volume from three face areas) individually, then compare working with a neighbour. Circulate to check that Q6 is approached via the product of dimensions, not by guessing.
38-50Introduce the think-and-reflect prompt: the stacked-cards model explains volume = area of base × height — students write two or three sentences justifying this. Selected responses are shared and the class weighs the book's two pool options, (A) 8 ft deep × 50 ft × 20 ft and (B) 6 ft deep × 100 ft × 15 ft (8000 ft³ against 9000 ft³), naming the parameters (depth, footprint, volume in litres) they consider (1 cubic foot ≈ 28.3 litres).

Homework

Exercise Set 14.1 Q2, p.125 (how many small cubes of side 20 cm fit in a 2 m box); Exercise Set 14.1 Q4, p.125 (two cubes joined end to end — surface area of resulting cuboid)

Assessment

Competency (C-5.2). Derives and applies surface-area and volume formulas for standard solids.

Learning outcome. Students can derive and apply the total surface area formula TSA = 2(wl + hl + hw) and volume formula V = lwh for a cuboid and their special-case forms 6a² and a³ for a cube.

A cube of side a cm and a cuboid of dimensions 2a cm × a cm × (a/2) cm are made from the same material. Which of the following statements correctly compares their total surface areas?

  • A The cube and the cuboid have equal total surface areas, since they are made from the same material.
  • B The cube has total surface area 6a² cm² and the cuboid has total surface area 6a² cm², so they are equal.
  • C The cube has total surface area 6a² cm² and the cuboid has total surface area 7a² cm², so the cuboid has the larger surface area.
  • D The cuboid has the smaller surface area because its longest dimension is 2a, which dominates.

C. The cube has total surface area 6a² cm² and the cuboid has total surface area 7a² cm², so the cuboid has the larger surface area.

What each wrong option reveals

  • A Confuses 'same material' with 'same surface area' — the two solids have the same volume (a × 2a × a/2 = a³ = volume of cube) but not the same surface area; being made from the same material is irrelevant.
  • B Applies the cube formula 6a² to the cuboid without substituting its actual dimensions into TSA = 2(wl + hl + hw); the cuboid's TSA = 2(2a·a + (a/2)·2a + (a/2)·a) = 2(2a² + a² + a²/2) = 2(3.5a²) = 7a².
  • D Reasons qualitatively from the longest dimension rather than computing all face areas; the longest dimension alone does not determine TSA.
Chapter 14 · 16 periods of 50 min

Also saved as 10 periods or 13 periods.

  • 2Surface Area and Volume of Cuboids — Ratio and Packing Problems
    Cuboids and Cubes
  • 3Right Circular Cylinder — Cross-Sections and Curved Surface Area
    Right Circular Cylinder
  • 4Volume of a Cylinder and the Stacking-Slices Justification
    Another Way of Understanding Why the Above Formulas Are True
  • 5Right Circular Cone — Slant Height and Curved Surface Area
    Cones
  • 6Volume of a Cone — One-Third Rule and Experimental Verification
    Cones
  • 7Pyramidal Shapes — Volume and Surface Area from Polygonal Bases
    Pyramidal Shapes
  • 8Spheres — Archimedes' Surface Area Formula via Cylinder Comparison
    Spheres and Hemispheres
  • 9Hands-On String-Winding Activity to Verify Sphere's Surface Area
    Hands-on Activity to Verify the Formula for Surface Area of a Sphere
  • 10Hemispheres — Curved and Total Surface Area, Volume
    Hemisphere
  • 11Areas and Volumes Around Us — Identifying and Estimating Real Solids
    Areas and Volumes Around Us
  • 12Guesstimate Problems — Modelling with Spheres and Cylinders
    Guesstimates: Did You Ever Wonder?
  • 13Consolidating Cylinder and Sphere Formulas Through Multi-Step Problems
    Right Circular Cylinder (Revisit)
  • 14Cone Surface Area and Volume in Applied and Rotational Problems
    Cones (Revisit)
  • 15Surface Area and Volume in Context — Guesstimate and Measurement Applications
    Guesstimates: Did You Ever Wonder? (Revisit)
  • 16Whole-Chapter Synthesis — Connecting Formulas, Methods and Models
    Synthesis
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