Lesson 13: Surface Area and Nets

A gift box needs wrapping paper. The box is not filled — it is covered, and the amount of paper depends on the area of every face added together: its surface area. Surface area decides how much cardboard a cereal box uses, how much paint a storage tank needs, and how much foil wraps a chocolate bar. In this lesson you will unfold boxes into flat nets, pair up their matching faces, and calculate surface area reliably — including the classic open-top box, which is missing a face on purpose.

Nets: unfolding a solid

A net is a flat pattern that folds up into a solid. A cube's net is six squares in a cross-like arrangement. A rectangular prism (a box) has six faces in three matching pairs: front and back are identical, left and right are identical, and top and bottom are identical. That pairing is the key to every surface-area calculation: find the area of one face from each pair, add the three areas, and double.

The surface-area formula in action

For a box with length l, width w and height h: Surface area = 2(lw + lh + wh) — each term is one pair of faces, doubled.

Worked example 1. A cube has edge 3 cm. Each face is 3 × 3 = 9 cm², and there are 6 faces: SA = 6 × 9 = 54 cm².

Worked example 2. A box measures 4 cm × 3 cm × 2 cm. Face pairs: 4 × 3 = 12, 4 × 2 = 8, 3 × 2 = 6. Sum: 12 + 8 + 6 = 26. Double: SA = 52 cm².

Worked example 3 (wrapping a gift). A gift box measures 8 in × 6 in × 4 in. Pairs: 8 × 6 = 48, 8 × 4 = 32, 6 × 4 = 24. Sum = 104. SA = 2 × 104 = 208 in² of paper — plus a little extra for overlaps and tape, which is why your estimate should always be rounded up in a shop.

The open-top box

An aquarium has no top — the fish need air (and food dropped in). Counting only five faces changes the method: compute the base once, then the four walls.

Worked example 4. An open-top tank measures 20 in long, 10 in wide, 12 in tall. Bottom: 20 × 10 = 200. Front and back: 2 × (20 × 12) = 480. Two ends: 2 × (10 × 12) = 240. Total glass: 200 + 480 + 240 = 920 in². Lids, baking trays, swimming pools (walls plus floor) and open crates all work this way: draw the net, cross out the missing face, and add what remains.

In the real world: cardboard and paint

Manufacturers count surface area by the tonne: a cereal box 20 cm × 6 cm × 28 cm uses 2 × (120 + 560 + 168) = 2 × 848 = 1,696 cm² of cardboard before flaps and glue tabs. Saving just 50 cm² per box across a million boxes saves 5,000 m² of card — which is why designers obsess over nets. Painters do the same sum for walls and tanks, subtracting doors and windows instead of adding lids.

Common mistakes

• Counting a face twice — or never. Every box has exactly six faces (three pairs). Listing them as pairs, then doubling the sum, makes a missed face nearly impossible.
• Confusing surface area with volume. Volume fills (l × w × h, cubic units); surface area covers (sum of face areas, square units). A gift needs paper by area, not by volume.
• Including the top of an open box. Read the question: "open-top", "without a lid" and "tank" are instructions to leave one face out.
• Forgetting to double. The sum 12 + 8 + 6 = 26 counts only three faces. The full surface area is 2 × 26 = 52 cm².

Exam Experience

1. What is the surface area of a cube with edge 2 cm? A) 8 cm² B) 16 cm² C) 24 cm² D) 48 cm²

2. What is the surface area of a rectangular prism measuring 5 cm × 4 cm × 3 cm? A) 60 cm² B) 94 cm² C) 47 cm² D) 120 cm²

3. How many faces does a triangular prism have? A) 4 B) 5 C) 6 D) 9

4. (Constructed response) A cereal box measures 20 cm tall, 28 cm wide and 6 cm deep. (a) Calculate the surface area of the box. Show how you paired the faces. (b) The manufacturer prints "10% extra cardboard is used for flaps and glue". How much cardboard is used for one finished box? (c) Explain why the company cares about surface area rather than volume when ordering cardboard.

Answer key and explanations

• 1. C — Each face is 2 × 2 = 4 cm²; 6 × 4 = 24 cm².
• 2. B — Face pairs: 5 × 4 = 20, 5 × 3 = 15, 4 × 3 = 12. SA = 2 × (20 + 15 + 12) = 2 × 47 = 94 cm².
• 3. B — A triangular prism has 2 triangular ends plus 3 rectangular sides = 5 faces.
• 4. (a) Pairs: 20 × 28 = 560, 20 × 6 = 120, 28 × 6 = 168. SA = 2 × (560 + 120 + 168) = 2 × 848 = 1,696 cm². (b) 10% of 1,696 = 169.6, so total = 1,696 + 169.6 = 1,865.6 cm². (c) Cardboard is a flat sheet material bought by area; the box is made by folding a net, so the sheet area needed is the surface area (plus flaps). Volume describes the storage space inside, not the material used.

Scoring guide for question 4 (4 points)

• 1 point for correctly identifying the three face pairs with areas 560, 120 and 168 cm².
• 1 point for the surface area, 1,696 cm², from doubling the sum.
• 1 point for the finished-box figure, 1,865.6 cm² (accept 1,866), from adding 10%.
• 1 point for the explanation linking cardboard (a sheet, bought by area) to surface area rather than volume.

Going further: surface area on the job

Worked example 5 (painting walls). A bedroom is 5 m long, 4 m wide and 3 m high. Painting the four walls only (not floor or ceiling): two walls are 5 × 3 = 15 m² each, two are 4 × 3 = 12 m² each. Total: 2 × 15 + 2 × 12 = 30 + 24 = 54 m² of paint coverage needed. Decorators do exactly this sum, then subtract doors and windows — surface area with pieces removed, just like the open-top box.

Worked example 6 (joined cubes). Two cubes of edge 3 cm are glued face-to-face. Separately they would need 2 × 54 = 108 cm² of covering. Joined, they form a 6 × 3 × 3 prism: face pairs 6 × 3 = 18, 6 × 3 = 18, 3 × 3 = 9, so SA = 2 × (18 + 18 + 9) = 90 cm². The two glued faces (9 cm² each) vanished from the outside — joining shapes always hides surface area, which is why ice cubes melt slower in a clump and radiators have fins to do the opposite.

Worked example 7. An open crate measures 30 cm × 20 cm × 15 cm (no lid). Base: 600 cm². Long walls: 2 × (30 × 15) = 900 cm². Short walls: 2 × (20 × 15) = 600 cm². Total timber: 600 + 900 + 600 = 2,100 cm². Five faces, carefully listed — listing beats memorising for anything open-topped.

Quick practice

• 1. Surface area of a cube with edge 10 cm.
• 2. Surface area of a box 2 cm × 2 cm × 5 cm.
• 3. How many faces does an open-top cube have?
• 4. Surface area of a box 7 cm × 3 cm × 2 cm.

Answers. 1. 6 × 100 = 600 cm². 2. Pairs: 4, 10, 10 → 2 × 24 = 48 cm². 3. Five (four walls plus the base). 4. Pairs: 21, 14, 6 → 2 × 41 = 82 cm².

Investigation: the roll of wrapping paper

A roll of wrapping paper is 50 cm wide and 300 cm long: a total area of 50 × 300 = 15,000 cm². Your gift boxes each measure 25 cm × 15 cm × 10 cm. Each face pair: 25 × 15 = 375, 25 × 10 = 250, 15 × 10 = 150. Surface area per box: 2 × (375 + 250 + 150) = 2 × 775 = 1,550 cm². How many boxes can the roll wrap? 15,000 ÷ 1,550 = 9.67…, and boxes are whole, so 9 boxes (using 9 × 1,550 = 13,950 cm², with 1,050 cm² left for patching mistakes). Two rounding lessons hide here: we rounded down because a tenth box cannot be wrapped in leftover scraps, and the true requirement is slightly more than the surface area anyway, since real wrapping overlaps edges. Estimation in the real world is not sloppy mathematics — it is exact mathematics plus an honest allowance for waste, stated out loud. Factories do this at scale: the allowance is called the wastage factor, and getting it wrong by 1% can cost more than the designer's salary.

Challenge corner

A door is 2 m tall and 0.9 m wide. One side has area 2 × 0.9 = 1.8 m², and painters must paint both sides: 3.6 m² in total (plus the thin edges, which professionals include and exam questions usually ignore — read each question to see which convention it uses). Doors, book covers and phone screens are all two-faced problems: the "surface" includes a back that diagrams rarely show. Second workout: a cube of edge 7 cm has surface area 6 × 7² = 6 × 49 = 294 cm². Compare its volume, 7³ = 343 cm³: as cubes grow, volume (cubed) outruns surface area (squared) — which is why big animals need clever lungs and why crushed ice melts faster than block ice. One formula pair, correctly handled, keeps explaining the world; that is the hallmark of mathematics worth learning.