Mathematics 11–12 · Year 11
Why A4 is 210 by 297: the surd sqrt 2 and index laws in the A-series paper sizes
Working with functions: Algebraic techniques (Mathematics Advanced, Year 11); Applications of measurement: Practicalities of measurement (Mathematics Standard, Year 11)
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The idea
A sheet that keeps its shape when folded in half must have sides in the ratio sqrt 2 : 1, and fixing A0 at one square metre makes every A-size an exact power of 2 that the index laws predict.
What you need
- Sheets of A3, A4 and A5 paper (or one A3 sheet folded)
- A metre rule reading to 1 mm, a kitchen or laboratory scale reading to 0.1 g
- The ISO 216 size table as published by the University of Cambridge Computer Laboratory page on international paper sizes
How to do it
- Measure the long and short sides of A3, A4 and A5 and compute each ratio long / short.
- Fold an A4 sheet in half across its long side and show the half has the same ratio; write the condition x / 1 = 1 / (x/2) and solve it to get x = sqrt 2.
- Given A0 has area 1 m^2 and ratio sqrt 2, show its sides are 2^(-1/4) m and 2^(1/4) m, and that A_n has sides 2^(-1/4 - n/2) m and 2^(1/4 - n/2) m.
- Evaluate the A4 sides from the index formula and compare with the measured 210 mm by 297 mm.
- Weigh 10 sheets of A4 printer paper, compute the mass per square metre and compare with the grams per square metre printed on the ream.
What you should see
The ratios of the ISO sizes are 420/297 = 1.414, 297/210 = 1.414 and 210/148 = 1.419, against sqrt 2 = 1.4142; the published sizes are A0 841 x 1189 mm, A3 297 x 420, A4 210 x 297 and A5 148 x 210. The index formula gives A0 as 840.9 by 1189.2 mm and A4 as 210.2 by 297.3 mm, which ISO 216 rounds to whole millimetres. A4 has area 1/16 m^2 (0.0624 m^2 after rounding), so 10 sheets of 80 g/m^2 paper weigh about 50 g. The learner knows it worked when the folded half keeps the same ratio to within a millimetre of measurement and the index formula reproduces the table.
What changes
- What you change
- the A-size number n (A3, A4, A5)
- What you measure
- the measured short and long side lengths, and the mass of 10 sheets
- What you keep the same
- the same metre rule read to 1 mm
- sheets from the same ream at one grammage
- each side measured between the outer edges of the cut
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A4 is half the size of A0; it is one sixteenth, because four halvings take A0 to A4.
- The ratio 297/210 is exactly sqrt 2; it is a rounded value of the irrational number 1.41421...
Safety card
Hazards
- paper cuts
Controls
- handle sheet edges with care
Note
No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Working with functions; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-01
- Mathematics Standard 11–12 Syllabus (2024), Year 11 focus area Applications of measurement; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-11-05
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.