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The Platonic Solids
Students learn about geometry by constructing models of 3D-‐
shapes. This is done with the aid of Polydron model sets. In
particular they look at regular polygons (2D shapes with equal
length sides) and create 3D shapes with these. Students
'discover' Euler's formula relating the number of edges, faces
and vertices (corners) of 3D shapes.
They examine models of the only five 3D shapes which can be
made from just one type of regular polygon. These are called
the Platonic Solids, one example is the cube as it is made
entirely of squares. Students experiment with making models
of these five shapes using straws and pipe cleaners. This
requires teamwork and problem solving. Archimedean solids,
3D shapes made from two types of regular polygon, can also
be created with the Polydron set. Eg. the football and the
carbon molecule called the Buckyball.
Equipment Overview
Equipment -‐ Kit
• Models of the 5 Platonic Solids.
– The Basic Polydron Platonic Solids Set works well.
– Or these can be made with card, using the nets.
• Polydron kits for constructing polyhedra. – We recommend the Frameworks Class Set.
– Two sets of the Frameworks Platonic Solids Set will cover the
basic activity (no Buckyball).
Equipment -‐ Consumables
• Straws
– About 6” long. – Allow 15 per student.
• Pipe Cleaners – Allow 15 per student.
– Ideally 2-‐3” long. Available as 12” long -‐ cut into quarters.
• Worksheets
• Printed nets of the Platonic Solids on card.
Equipment -‐ Stand
• Table
• Laminated Construction Instructions
• Laminated Properties Table
• Poster + Poster board/wall
• Examples of irregular polygons
• Examples of other polyhedra Eg. Prism (Toblerone)
Equipment -‐ Costs ITEM PRICE
Polydron Platonic Solid Set 26.00
Polydron Frameworks Class Set 132.00
Polydron Frameworks Platonic Solid Set 17.00
Short ArtStraws (215) 4.50
12” Pipe clearers (200) 6.35
Prices not provided for paper, card, ink etc. Each institution will have its own Office Supplies provider or access to these supplies.
All of the above are available from Evans Educational: www.evanseducational.ie Contact Noreen: [email protected]
Alternatively available directly from Polydron: www.polydron.co.uk or www.amazon.co.uk
Preparation
• Print nets on coloured card
• Print instruction sheets + laminate if possible
• Print Worksheets
• Cut pipe cleaners into quarters
• Practice making the models
• Make the solid models for show
• Make sure you know the correct numbers for the faces, vertices
and edges of each solid, and the names!
Demonstration
This activity examines each of the following:
• Polygons
• Polyhedra
• Euler’s Formula
• Platonic Solids
• Constructions
• Archimedean Solids (extension)
Polygons
• Ask if they know what a polygon is:
– It is a flat (2-‐D) shape made of straight lines.
• Ask if they can think of a shape that’s not a polygon.
– No curves allowed -‐ so a circle is NOT a polygon.
• Have examples to hand: rectangle, any triangle,
hexagon, any shape drawn with a ruler!
Regular Polygons
• What does it mean if a polygon is regular?
– All sides are equal length.
• Square is a regular quadrilateral
• Equilateral triangle is a regular triangle.
• Ask if they know of other examples.
• Use the Polydron shapes to show these.
Polyhedra
• What is a polyhedron?
– A 3D shape where each face is a polygon.
• Examples include prisms and cuboids.
• Show them using a toblerone box and a book.
• Can they think of a shape that is not a polyhedron? – No curves: The cone & sphere are NOT polyhedra.
Euler’s Formula
• Ask them to count the number of vertices and
edges on each display model and fill the numbers in
on their worksheets.
• Ask them to perform the F+V-‐E calculation.
– They should see it’s always 2.
• Explain this is Euler’s formula
– It is true for any (convex) polyhedron.
Platonic Solids -‐ Discovery
• Ask them what’s special about these polyhedra (the
models of the Platonic solids).
– What have they got in common?
– What shapes are they made of?
– Is it any triangle, or are they special?
• Lead them to discover that each model is made of only
one shape. And that shape is a regular polygon.
Platonic Solids -‐ Names
• Ask them to name these polyhedra.
• For tetrahedron they may say ‘pyramid’
– Explain that triangular pyramid is correct.
– Square pyramids are the ones we see in Egypt and they are
half of the octahedron.
• Let them build Platonic solids with the Polydron pieces
based on the number of faces.
• The two larger shapes may take two people to build.
Platonic Solids -‐ Symmetry
• Ask about how symmetric the shapes are, what kind of
symmetry they have.
– They should see that every vertex is the same!
• What’s special about these 5 polyhedra is:
– They are made only of one type of regular polygon.
– The same number of polygons meet at every vertex.
• Any (convex) polyhedron that meets those
requirements is called a Platonic Solid.
Platonic Solids -‐ Details Shape Name Faces Faces /
Vertex Vertices Edges
Triangle Tetrahedron 4 3 4 6 Triangle Octahedron 8 4 6 12 Triangle Icosahedron 20 5 12 30 Square Cube 6 3 8 12 Pentagon Dodecahedron 12 3 20 30
There are only these 5 Platonic Solids! Why can’t there be any more?
Construction-‐ Methods There are three methods of construction: • The first uses whole shapes that click/stick together, these
are made with the Polydron Sets. • The second uses ‘edges’ and ‘vertices’ that join together to
make a skeleton of the solid. In this case the edges are straws and the pipe cleaners join the straws at the vertices.
• The third method is to cut a ‘net’ out of card and fold it into the shape. The net is a ‘flattened out’ version of the shape.
Students should be able to perform the first construction quickly. The second construction is more of a challenge. The third construction is a simple take-‐home task.
The easiest way to construct the Platonic Solids is to focus on the number of faces that meet at each vertex. Alternatively, it can be done by knowing the number of faces in the shape.
The smallest three shapes (the tetrahedron, the cube and the octahedron) are easy to make by looking at the pictures or at the display models. When making the shapes with the Polydron Sets the dodecahedron is ok too.
For the icosahedron it makes sense to make the ‘top’ and ‘bottom’ vertices from 5 triangles each, and then use 10 triangles in alternate order (point up, point down…) to join the top and bottom. This strategy works well for both construction methods.
For the dodecahedron it works best to make two pentagons – with an extra edge at each vertex. Then have someone hold these while the 10 connecting pentagons are made with the remaining 10 edges, alternating joining to an ‘upper’ or ‘lower’ protruding edge.
Stability
The Vertex+Edge made dodecahedron is very unstable, especially when made with materials which allow a lot of flexibility either in the edges or vertices. It can also be made with edges of half the length. This will improve the stability but not fix the problem. Of the Platonic solids its polygons have the largest angles and the least inherent stability, next is the cube. The shapes made with triangles are much stronger and will stay in the correct positioning even if ‘nudged’. This is important for engineering applications.
