What it shows
The string that covers the curved surface of a hemisphere covers exactly two flat circles of the same radius, so a hemisphere has an area 2 pi r squared. A whole sphere is therefore four circles, 4 pi r squared.
The science
Archimedes found that the curved surface of a sphere equals the area of four great circles, 4 pi r squared. A thread wound tightly on the surface covers it with the same density at any place on it, so the length of thread needed measures the area. The flat circle of radius r is covered by the same thickness of thread in a spiral, so if the hemisphere needs twice the length of thread that the circle needs, the area is twice, 2 pi r squared.
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.
- A foam or polystyrene ball, 8 to 10 cm across, or a rubber ball and clay
- A long thin cotton thread, about 10 m
- Strong glue or double-sided tape
- A pencil, a pin and a ruler
- Thick cardboard
- Scissors
- A knife for an adult
Safety
For this project: An adult must cut the foam ball in half with a knife. Use the knife on a flat board and cut away from the body. Keep pins away from small children.
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.
An adult cuts the foam ball into two halves, and you keep one hemisphere. Measure its radius r across the flat face.
Fix one end of the thread with a pin at the top pole of the curved surface and glue the thread down spiralling outward around the pole in neat touching turns, until the whole curved surface is covered. Do not let turns overlap.
Mark and cut the thread at the point where the surface is covered, and unwind it. Measure its length L1.
Draw a circle of the same radius r on cardboard and cut it out. Glue thread on in a tight spiral from the centre out to the edge in the same way and cut it.
Measure the length L2 of thread that covered the flat circle, and calculate L1 divided by L2.
The ratio should be close to 2. Write the result and the percentage error.
Compute the sphere area 4 pi r squared and the hemisphere curved area 2 pi r squared for your radius, and compare with the thread measurements.
Show the ratio using a second hemisphere, so that visitors see that the whole sphere needs four circles of thread.
What you should see
The thread on the hemisphere is about twice as long as that on the flat circle, for example 6.4 m against 3.2 m, supporting that the curved area is 2 pi r squared.
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 the surface area of a sphere?
4 pi r squared.
2What is the curved surface area of a hemisphere?
2 pi r squared, and the total surface with the flat face is 3 pi r squared.
3Why can thread measure area?
The same thickness of thread covers equal areas with equal lengths.
4Who first proved the formula?
Archimedes, about 2200 years ago.
Take it further
Use the same thread method to estimate the surface area of an orange and compare with the formula from its measured diameter.
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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