About this experiment
Rotate a solid and mark each surface, edge or vertex as you find it. Separate flat faces to uncover hidden features and compare polyhedral counts with the different conventions needed for curved solids.
Learning goals
- Identify flat faces, curved surfaces, edges and vertices.
- Count hidden as well as front-facing features.
- Apply Euler counting only to the convex polyhedra in this collection.
How to use it
- 1Count cube surfaces by clicking each one once and rotating to the back.
- 2Switch to edges and vertices and use the numbered buttons if a feature is hard to select.
- 3Separate the flat faces of a pyramid, then reveal its counts.
- 4Compare a cylinder, cone and sphere using the stated curved-surface conventions.
The model
Assumptions
- Polyhedra are convex closed solids. Their model edges come from shared polygon boundaries, not from rendered mesh triangulation.
- Flat faces and curved surfaces are counted separately. Cylinder circles count as curved edges; a sphere has one curved surface and no edges or vertices.
- Euler counting is not applied to the curved-surface conventions. Visual face separation and size changes leave the original feature counts unchanged.
Try this in class
- Q1Why must a cube vertex hidden behind the front faces still be counted?
- Q2Why should the triangles used to draw a sphere not be counted as its mathematical faces?
Frequently asked questions
Does a curved solid have polygon faces in this activity?
No. The activity explicitly separates flat faces from curved surfaces so different classroom conventions are not silently mixed.
Can the same feature be counted twice?
Clicking a marked feature removes its mark. Each model feature has one index, so the progress count uses distinct features.