Why A4 is the shape it is
Tonight's idea
Most countries print on paper called A4. It is 210 mm wide and 297 mm long. Those seem odd numbers to choose. There is a reason.
Cut A4 in half across its long side. Each half is 210 mm by 148.5 mm. This size is called A5. It looks like A4, only smaller.
"Looks like" needs a test. Divide the long side by the short side. Call the answer the shape number. Rectangles with the same shape number are the same shape.
A4: \(297 \div 210 \approx 1.414\). A5: \(210 \div 148.5 \approx 1.414\). The sign \(\approx\) means "is about". Halve A5 and you get A6, the same shape again.
Shape number = long side ÷ short side. For A paper it is about 1.414, before and after the cut.
Use a calculator for the divisions on this sheet.
Watch one: a card that fails
A card is 300 mm by 200 mm. Its shape number is \(300 \div 200 = 1.5\).
Halve the long side. The card is now 200 by 150, and \(200 \div 150 \approx 1.33\). It got stubbier. Halve again: 150 by 100, and \(150 \div 100 = 1.5\). It flips between two shapes.
A square does worse. Its shape number is 1, and half a square has shape number 2. Only a shape number near 1.414 survives the cut.
Your turn
- Printers round A5 to 148 mm by 210 mm. Cut it in half across the long side to make A6. (a) What are the sides of A6? (b) Find its shape number to two decimal places.
- Lay two A4 sheets side by side with their long edges touching. This is A3. (a) What are its sides? (b) Find its shape number. (c) How many A6 postcards can you cut from one A3 sheet?
- Letter paper, used in the USA, is 216 mm by 279 mm. (a) Find its shape number to two decimal places. (b) Halve it across the long side. Find the new sides and the new shape number. (c) Is half a Letter sheet the same shape as Letter?
- The biggest sheet, A0, has an area of 1 square metre. Halving it gives A1, then A2, A3 and A4. (a) How many A4 sheets make one A0? (b) One square metre is 10,000 cm². What is the area of one A4 sheet? (c) A4 is 21 cm by 29.7 cm. Multiply to check. Why might the two answers differ a little?
Stretch: where 1.414 comes from
The 1.5 card is less than twice as long as it is wide. For any such rectangle, halving turns its shape number into 2 ÷ that number. Check this with the card. But now halve a 300 mm by 100 mm card, whose shape number is 3. What is its new shape number, and why can no number greater than 2 survive unchanged? So the special number must be between 1 and 2. There it equals 2 ÷ itself, so its square is 2. Try \(1.41 \times 1.41\) and \(1.42 \times 1.42\). What do you find?
Answers
- (a) Half of 210 is 105, so A6 is 105 mm by 148 mm. (b) \(148 \div 105 \approx 1.41\).
- (a) The short sides join up: \(210 + 210 = 420\). A3 is 297 mm by 420 mm. (b) \(420 \div 297 \approx 1.414\). (c) A3 is 2 A4s, 4 A5s, or 8 A6s. So 8 postcards.
- (a) \(279 \div 216 \approx 1.29\). (b) Half of 279 is 139.5, so the piece is 139.5 mm by 216 mm. \(216 \div 139.5 \approx 1.55\). (c) No. The shape numbers 1.29 and 1.55 are different, so the half sheet is longer and thinner.
- (a) Each cut doubles the count: 2, 4, 8, 16. Sixteen A4 sheets. (b) \(10{,}000 \div 16 = 625\) cm². (c) \(21 \times 29.7 = 623.7\) cm². The sides of real paper are rounded to whole millimetres, so the answers are close but not equal.
Stretch. \(2 \div 1.5 \approx 1.33\), which matches the first card. The 300 by 100 card halves to 150 by 100, with shape number 1.5, not \(2 \div 3\). If the old shape number is greater than 2, half the long side is still longer than the short side, so halving simply halves the shape number; it cannot stay unchanged. A shape that survives must be between 1 and 2. In that range, the old short side becomes the new long side, and the new short side is half the old long side. So the special number must equal 2 ÷ itself, and its square is 2. \(1.41 \times 1.41 = 1.9881\) is just under 2, and \(1.42 \times 1.42 = 2.0164\) is just over. The special number lies between them: \(1.414 \times 1.414 = 1.999396\).