Lesson 18: Surface Area

§Text Passage

1Summary 重點摘要

Surface area = all faces added up. Cuboid = 2×(lw + lh + wh). Learn to calculate cuboid and cube surface area.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Cuboid Surface Area FormulaThe surface area of a cuboid follows SA = 2 × (lw + lh + wh). For example, with length 5 cm, width 3 cm and height 2 cm, SA = 2 × (15 + 10 + 6) = 2 × 31 = 62 cm². Why multiply by 2? Because a cuboid has 6 faces in three pairs — top and bottom, front and back, left and right — and the two faces of each pair are the same size, so the three pairs are each counted once and then doubled, covering all six faces. A common error is dropping one term inside the brackets, or forgetting the × 2 and getting half; write the formula first, match length, width and height term by term, and substitute step by step.
2. Block 2: Face by Face (Easy)The face-by-face method is easier to understand: top = bottom = length × width = 5 × 3 = 15 cm², two faces making 30 cm²; front = back = length × height = 5 × 2 = 10 cm², two faces making 20 cm²; left = right = width × height = 3 × 2 = 6 cm², two faces making 12 cm²; in total 30 + 20 + 12 = 62 cm². Why does the face-by-face method suit beginners? Because it lays the six faces out pair by pair, matching the thinking of nets, with every step visible and no face easily missed. A common error is mismatching a pair's dimensions, such as treating the front as 3 × 2; write the dimensions of each pair first, then calculate, and the pairing stays ordered.
3. Block 3: Cube Surface AreaThe surface area of a cube follows SA = 6 × side². For example, with side 4 cm, one face is 4 × 4 = 16 cm², and six faces give 6 × 16 = 96 cm². Why so simple? Because all six faces of a cube are identical squares, so one face times 6 finishes the job, far less work than a cuboid. In the formula, side² means the side multiplied by itself, not the side × 2. A common error is computing 4² as 8 and getting 6 × 8 = 48 cm²; remember that a square is self-multiplication, 4² = 16, and never omit the square unit cm², since an area answer without its unit is incomplete.
4. Block 4: Volume vs Surface AreaVolume and surface area are two different things: volume tells how much space a solid takes, in cm³, while surface area tells how large the outside is, in cm². For the same solid the two values usually differ. For a box of 5 by 3 by 2, the volume = 5 × 3 × 2 = 30 cm³, while the surface area is 62 cm². Why tell them apart? Because word problems ask both: how much it holds calls for volume, how much paper wraps it calls for area. A common error is mixing the units, writing cm² for a volume or cm³ for an area; read the question first to see whether it asks about space or surface, then choose the formula and the unit.
5. Block 5: Word Problem: Wrapping PaperWord problem: a gift box is 20 cm long, 15 cm wide and 10 cm high; how much wrapping paper is needed? This asks for the surface area: SA = 2 × (20×15 + 20×10 + 15×10) = 2 × (300 + 200 + 150) = 2 × 650 = 1300 cm². Why answer wrapping with surface area? Because the paper covers the box's entire outer surface, and the amount needed corresponds exactly to the total of the six faces; real wrapping adds a margin for the seams, but the mathematics counts the six faces. A common error is counting only the four sides and dropping the top and bottom; be clear that the paper covers all six faces and check the pairs one by one.
6. Block 6: Word Problem: Painting RoomWord problem: a room is 4 m long, 3 m wide and 2.5 m high, and four walls plus the ceiling are to be painted, with the floor left alone. Area = four walls + ceiling = 2×(4×2.5 + 3×2.5) + 4×3 = 2×(10 + 7.5) + 12 = 35 + 12 = 47 m². Why not apply the full formula? Because the painting question excludes the floor, so the bottom face must be removed from the six — a flexible variation of surface area; the four walls are the front-and-back pair plus the left-and-right pair, and the ceiling is length × width. A common error is counting all six faces, adding the floor's 12 m² by mistake; circle which faces are to be painted, then compute face by face.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Surface area = all faces added up
• Cuboid: 2×(lw + lh + wh)
• Cube: 6 × side²
• Unit: cm² (same as area)
• Volume: cm³, surface area: cm²