What it shows
Only five solids can be made with every face the same regular polygon and the same number of faces at every corner: the tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons) and icosahedron (20 triangles). Each is folded from a flat net.
The science
At a corner of a solid the angles of the faces meeting there must add to less than 360 degrees, or the surface would lie flat. Equilateral triangles have 60 degree corners, so 3, 4 or 5 can meet (180, 240, 300 degrees) but not 6. Squares have 90 degree corners, so only 3 can meet. Regular pentagons have 108 degree corners, so only 3 can meet. Hexagons have 120 degree corners and three of them make 360, which is flat. This leaves exactly five solids.
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.
- Cereal-box cardboard or thick chart paper
- A ruler, compass and protractor
- A blunt knife or an old pen for scoring folds
- Glue or tape
- Paints in five colours
Safety
For this project: An adult must help with scoring and cutting.
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.
Step 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.
Draw an equilateral triangle of side 5 cm with the compass and use it as a template for all the triangle nets.
Draw the tetrahedron net: a large triangle divided into four small ones. Score the inner lines, fold and glue the tabs.
Draw the cube net as a cross of six 5 cm squares and fold it up.
Draw the octahedron net as a strip of six triangles in a row, pointing alternately up and down, with one extra triangle above the second and one below the third. Add tabs on the free edges, curl the strip into a ring and close the two ends with the extra triangles.
Draw a regular pentagon of side 4 cm using 108 degree corners, and make the dodecahedron net as two flowers of six pentagons each.
Draw the icosahedron net as a strip of ten triangles with five more above and five below, then fold it round.
Count the vertices, edges and faces of each solid and record them in a table.
Paint each solid a different colour and label it with its name and the number of faces.
What you should see
Five finished solids with 4, 6, 8, 12 and 20 faces, and a table showing that for each one vertices minus edges plus faces equals 2.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01Why can only three squares meet at a corner?
Four squares would give 360 degrees, which lies flat and cannot close up.
02How many edges does a cube have?
12.
03Which solid has 12 faces?
The dodecahedron, made of regular pentagons.
04How many vertices does an icosahedron have?
12, with five triangles meeting at each.
Take it further
Make the cube and octahedron so one fits inside the other with vertices touching face centres: they are dual solids.
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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