Mathematics 7–10 · Year 8
Pythagoras by measurement: squares on dotty paper and a 3-4-5 rope
Measurement and space
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The idea
The square on the hypotenuse has the same area as the two squares on the other sides, which can be counted on dotty paper and checked with a tape measure on a knotted rope that forms a right angle.
Safety card
Hazards
- trip hazard from rope and tape on the ground
- sun exposure outdoors
Controls
- keep the rope area clear of other groups
- hats and shade
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's excursion and playground supervision rules.
What you need
- 1 cm dotty grid paper
- Ruler and set square
- 5 m of cord per group with 13 knots tied at 30.0 cm intervals, giving 12 equal spaces of 30.0 cm, and 3 pegs or three learners (the 12 spaces take 3.60 m and the 13 knots and two tails take the rest, so 4 m is not enough)
- 5 m tape measure
- Playground chalk for step 6
How to do it
- On dotty paper draw a tilted square whose side goes 3 across and 1 up; find its area by boxing it in a 4 by 4 square and subtracting four triangles (16 − 4 × 1.5 = 10). Repeat for sides 3 across 2 up, and 4 across 3 up.
- Tabulate across, up and area, and test the rule area = across² + up²; then note the square's side is the hypotenuse of the right-angled triangle with those legs.
- Take the knotted rope outside. Hold the first and thirteenth knots together at one corner, the fourth knot at the second corner (3 spaces along) and the eighth knot at the third corner (4 spaces further), and pull taut so the last 5 spaces run back to the start; check the corner at the fourth knot with a set square.
- Measure the three sides of the rope triangle with the tape (expected 90.0 cm, 120.0 cm, 150.0 cm) and compute 90² + 120² and 150².
- Use the rope to test whether a room corner, a doorway or a sports court corner is square, and record the discrepancy.
- Draw two more right-angled triangles with legs 60 cm and 80 cm, and 50 cm and 120 cm, on the playground with chalk, measure the hypotenuse and compare with the computed value.
What you should see
Tilted squares give areas 10, 13 and 25 for (3, 1), (3, 2) and (4, 3), each equal to across² + up². The rope triangle measures 90, 120 and 150 cm within about ±1 cm, and 90² + 120² = 8100 + 14 400 = 22 500 = 150². The chalk triangles give hypotenuses of 100.0 cm and 130.0 cm, measured within ±1 cm. A room corner that is square gives a rope loop that closes without slack; a corner 2 degrees out of square produces about 2.5 cm of mismatch over a 150 cm hypotenuse (by the cosine rule, 90 and 120 cm legs at 88 degrees give 147.5 cm). The learner knows it worked when the measured hypotenuse agrees with the computed one within 1 per cent.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The hypotenuse equals the sum of the other two sides.
- The theorem works for any triangle.
- a² + b² = c² means a + b = c after taking square roots.
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 K–10 Syllabus (2022), Stage 4 Right-angled triangles (Pythagoras' theorem); page read 2026-09-22MA4-PYT-C-01
- Australian Curriculum v9AC9M8M06
Sources
The pages the author read to write this activity.