What it shows
A closed loop of thread of fixed length can be made into a triangle, rectangle, square, hexagon or circle. The enclosed area grows as the shape gets more regular and is largest for the circle, so for a fixed perimeter the area can be very different.
The science
The perimeter of every shape here is 40 cm, but the area is not the same. For a rectangle with sides l and b, area = l times b, and it is greatest when l = b, so the square, with area 100 square cm, beats every other rectangle, such as 5 by 15, with area 75. Among all shapes with the same perimeter, the circle encloses the most area, the isoperimetric property. For a perimeter of 40 cm the circle has a radius of 6.37 cm and an area of about 127 square cm.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A thread loop with a total length of exactly 40 cm
- Squared paper with 1 cm squares, a few sheets
- Pins or small pieces of tape
- A ruler, a pencil and an eraser
- A thick card sheet to lay the thread on
- A calculator
Safety
For this project: An adult should supervise the use of pins, or use tape instead.
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
8 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.
Make a loop from thread by tying it so that the total length around is exactly 40 cm. Check with a ruler.
Lay the loop on squared paper as a rectangle 5 cm by 15 cm and use pins or tape to hold the corners. Count the squares inside and note the area, 75.
Reshape it into rectangles of 8 by 12 and 10 by 10 and count the squares again: 96 and 100.
Make an equilateral triangle of side about 13.3 cm and count squares, about 77.
Make a regular hexagon of side 6.7 cm, about 116 square cm, and an approximate circle by letting the loop form its own shape on a flat surface by gently pulling it with a pencil from the inside, about 127.
Count squares by marking each full square and adding half the number of part squares along the edge.
Compare the counted areas with the formulas: l times b, root 3 over 4 times a squared, 3 root 3 over 2 times a squared, and pi r squared.
Draw a bar chart ordering the areas from the smallest to the largest and write the conclusion.
What you should see
The areas come to about 75, 96, 100, 77, 116 and 127 square cm for the six shapes, and the circle, with the most sides, encloses the most.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is perimeter and what is area?
Perimeter is the length around the shape and area is the surface inside.
2Which rectangle with a perimeter of 40 cm has the greatest area?
The square of side 10 cm.
3Why is a circle best?
It is the shape that encloses the largest area for a given perimeter.
4Where do we use this idea?
In round water tanks and cans that need less material for a given volume.
Take it further
Fence a field with 100 m of rope against a wall on one side and find the rectangle with the largest area.
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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