What it shows
A rope with 13 knots at equal spacing is pulled into a triangle with sides of 3, 4 and 5 spaces. The angle between the 3 and 4 sides is exactly a right angle, because 3 squared plus 4 squared equals 5 squared, a trick used to lay out buildings for thousands of years.
The science
The converse of Pythagoras says that if the sides of a triangle satisfy a squared plus b squared equals c squared, the angle opposite c is a right angle. Since 9 + 16 = 25, any triangle with sides in the ratio 3 : 4 : 5 is right-angled. Equal knot spacing turns the rope into a measuring tool with no ruler, which is why Egyptian and Indian surveyors used rope stretchers to set out square corners for temples and fields.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- About 4 metres of thin rope or thick string
- A measuring tape
- Three pegs or three friends
- A set square to check
- Chalk to mark the ground
Safety
For this project: Peg the rope outdoors on soft ground, or hold it with fingers; never pull it tight near anyone's face.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
7 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Measure and tie 13 knots along the rope exactly 25 cm apart, so there are 12 equal spaces. Equal spacing is the whole trick; measure each gap from knot to knot.
Tie the two end knots together so the rope forms a loop of 12 spaces.
Peg one knot at a corner mark on the ground. Count 3 spaces along and peg that knot, pulling the side straight.
From the corner go the other way 4 spaces and peg that knot so all three sides are tight: 3, 4 and 5 spaces.
Check the corner with a set square or the corner of a book: it is a right angle.
Mark the two sides with chalk and extend them to lay out a square 2 metres on a side; check by measuring both diagonals, which should be equal at about 2.83 metres.
Try other knot counts like 5, 12, 13 with a longer rope of 30 spaces and test the corner again.
What you should see
The 3-4-5 corner matches a set square exactly, and the square laid out from it has equal diagonals, proving the corners are right angles.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why does 3, 4, 5 give a right angle?
Because 3 squared plus 4 squared is 9 + 16 = 25 = 5 squared, so by the converse of Pythagoras the angle opposite the 5 is 90 degrees.
2Name another set of three whole numbers that works.
5, 12, 13 or 8, 15, 17 or 6, 8, 10.
3Why check the diagonals of the square?
A four-sided shape with equal sides and equal diagonals must be a square with right angles.
4How many knots do you need for 12 spaces?
13 knots, since there is one more knot than spaces.
Take it further
Use the rope to check whether the corners of your classroom are true right angles and report the error in centimetres.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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