Math Detours · Grade 7 · Sharpen · 12

Wrapping a box

Surface area is the paper needed to cover a solid. Unfold the box into its net and it is six rectangles in three matching pairs.

Name ____________________Date ____________

Tonight's idea

Volume tells you how much a box holds. Surface area tells you how much paper covers it: the total area of its faces.

A box has six faces, hard to see all at once. So cut along some edges and fold it out flat. The flat shape is a net, and every face in it is a rectangle.

423 4 × 3 = 124 × 3 = 124 × 2 = 84 × 2 = 866
A 4 by 3 by 2 box and its net. Matching colours are matching faces.

Top and bottom match. So do front and back, and so do the two ends. Six faces, but only three different rectangles, each used twice. For this box: \(12 + 12 + 8 + 8 + 6 + 6 = 52\) square units.

For a box with length \(l\), width \(w\) and height \(h\): surface area \(= 2 \times (l \times w + l \times h + w \times h)\).

The formula just says: three rectangles, twice each. Area is counted in squares, so the units are cm², not cm³.

More nets: Eleven ways to unfold a cube.

Watch one: a shoebox with no lid

A shoebox is 30 cm long, 20 cm wide and 10 cm high.

The three rectangles: \(30 \times 20 = 600\), \(30 \times 10 = 300\) and \(20 \times 10 = 200\).

All six faces: \(2 \times (600 + 300 + 200) = 2200\) cm².

Without the lid, one \(30 \times 20\) face is missing: \(2200 - 600 = 1600\) cm².

With a face missing, the formula does not fit. Counting faces still works.

Your turn

  1. Find the surface area of (a) a box 5 cm by 4 cm by 3 cm; (b) a cube with edges of 3 cm.
  2. A cereal box is 30 cm tall, 20 cm wide and 7 cm deep. (a) Sketch its net and write the area on each face. (b) How much card does it use, ignoring flaps?
  3. A cube has surface area 150 cm². (a) What is the area of one face? (b) How long is each edge? (c) What is its volume?
  4. You have 24 small cubes, each 1 cm along every edge. You can stack them as a box 2 by 3 by 4, or as a tower 1 by 1 by 24. (a) Both hold 24 cm³. Find the surface area of each. (b) Explain why the tower needs so much more paper. Think about which small faces are hidden.

Stretch: three faces, no edges

You are told only the areas of three faces of a box that meet at one corner: 12 cm², 8 cm² and 6 cm². Find its surface area. Then find its volume, without guessing the edges. Hint: multiply the three areas together and see what you have made.

Answers

  1. (a) \(2 \times (20 + 15 + 12) = 94\) cm². (b) Six squares of \(3 \times 3 = 9\): 54 cm².
  2. (a) Two faces of \(30 \times 20 = 600\), two of \(30 \times 7 = 210\), two of \(20 \times 7 = 140\). (b) \(2 \times (600 + 210 + 140) = 1900\) cm².
  3. (a) \(150 \div 6 = 25\) cm². (b) 5 cm, since \(5 \times 5 = 25\). (c) \(5 \times 5 \times 5 = 125\) cm³.
  4. (a) Box: \(2 \times (6 + 8 + 12) = 52\) cm². Tower: \(2 \times (1 + 24 + 24) = 98\) cm². (b) The 24 cubes have \(24 \times 6 = 144\) small faces in total. In the tower each cube hides only the faces touching the cubes above and below it. Packed into the box, cubes touch on several sides, so more faces are hidden inside and fewer need covering.

Stretch. Surface area: \(2 \times (12 + 8 + 6) = 52\) cm². For the volume, the three areas are \(l \times w\), \(l \times h\) and \(w \times h\). Multiplied together they use each edge twice, so the product is volume × volume. \(12 \times 8 \times 6 = 576 = 24 \times 24\), so the volume is 24 cm³. The box is 4 by 3 by 2.

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