Today we used inspiration from Nrich and mathsisfun.com to explore the Platonic solids on Polypad.
Task 1 – Dodecahedron
Go to https://polypad.amplify.com/p#polygons and make a net of a dodecahedron. You can keep adding pentagons together and then highlight them all and click Fold Net.
If it is not quite right, click Unfold Net and then add/remove pentagons and try again.
Task 2 – Angle deficit
The angle deficit at a vertex of a polyhedron is a measure of how far short each angle sum is from 360°. For example, in a dodecahedron, three pentagons with interior angles of 108° meet at each vertex, so the angle sum is 324° and the angle deficit is 36°:
The total angle deficit is the sum of the deficits at each vertex.
Can you fill in the table for the Dodecahedron:
Platonic Solids
| Regular polygons | Faces | Vertex form | Angle Sum | Angle Deficit | Number of Vertices | Total Angle Deficit | |
| Cube | Squares | ||||||
| Tetrahedron | Triangle | ||||||
| Octahedron | Triangle | ||||||
| Icosahedron | Triangle | ||||||
| Dodecahedron | Pentagon | 5,5,5 | 324 | 36 |
Task 3 – Make the nets and fill in the table for the other four Platonic Solids. These are 3D shapes where the faces are congruent regular polygons and the same number of faces meet at each vertex.
Task 4 – Find all the Archimedean Solids which are formed by two or more types of regular polygons, each with the same side length and such that each vertex has the same pattern of polygons around it.
Further reading
https://polypad.amplify.com/lesson/five-platonic-solids
https://www.maths.ed.ac.uk/sites/default/files/atoms/files/platonic_solids_reduced.pdf