What it shows
A cone, a hemisphere and a cylinder with the same radius, and height equal to the radius for the cone and cylinder, have volumes in the ratio 1 to 2 to 3. Pouring rice from a cone and a hemisphere together exactly fills the cylinder.
The science
For radius r and height r, the cone has volume 1/3 pi r squared times r = pi r cubed over 3, the hemisphere has 2/3 pi r cubed, and the cylinder has pi r squared times r = pi r cubed. So the three are in the ratio 1/3 to 2/3 to 1, that is 1 to 2 to 3. The cone and the hemisphere together equal the cylinder. Archimedes valued this result so much that he asked for it to be marked on his tomb.
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.
- Thick card or thin plastic sheets
- A compass, ruler and scissors
- Clear tape
- A half rubber ball or a round steel katori of about 8 cm width as the hemisphere
- Rice or sand, about 500 g
- A plate or tray
- Paper to make a funnel
Safety
For this project: An adult must cut thick card or plastic. Do not let children put rice in their mouths. Clean up spills to avoid slipping.
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.
Choose the radius r, for example 4 cm, and measure the inside width of the hemispherical bowl so that it is 8 cm across and 4 cm deep.
Draw and cut a cardboard sector for the cone: with a slant height of about 5.7 cm, which is the square root of r squared + h squared with r = h = 4, and roll it into a cone with an open top of radius 4 cm. Tape its seam.
Make a cylinder of radius 4 cm and height 4 cm from card, with a base fixed on with tape. Tape all seams so that rice cannot leak.
Fill the cone with rice level with its rim, using a straight ruler to level it. Check that the height is also exactly 4 cm.
Pour the rice from the cone into the cylinder. It fills one third of the height.
Fill the hemisphere and pour it in. The cylinder is now filled to the top, since the hemisphere is twice the cone.
Empty the cylinder and fill the hemisphere to the brim, then pour it into the empty cone repeatedly to see that it takes two cones of rice to match one hemisphere.
Write the formulas and a table showing the three volumes in cubic cm for r = 4, which are about 67, 134 and 201.
What you should see
One cone of rice plus one hemisphere of rice exactly fills the cylinder, and the volumes of 67, 134 and 201 cubic cm for r = 4 agree with the pour test.
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 volume of a cone?
One third of pi r squared h.
2What is the volume of a sphere?
Four thirds of pi r cubed, so the hemisphere has two thirds of pi r cubed.
3What is the ratio of volumes of cone, hemisphere and cylinder here?
1 to 2 to 3.
4Why must the height of the cone equal its radius?
The ratio 1 to 2 to 3 holds only when height equals radius, and the hemisphere height is equal to its radius.
Take it further
Change the cone height to 2r and show that the volume doubles, then compare it with the hemisphere.
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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