🔍 Patterns in Prisms & Antiprisms
Explore fascinating mathematical patterns by building prisms and antiprisms of different sizes! A prism has two identical polygon bases connected by rectangles. An antiprism has two identical polygon bases connected by triangles, with one base twisted. Can you discover the hidden formulas that predict how many vertices (corners), edges, and faces(flat sides) these shapes will have?
🔢 Prism Pattern Investigation
Click the Build button to create each shape, then count the vertices (V), edges (E), and faces (F). Can you find a pattern?
| Build Shape | Base Shape | Sides (n) | Vertices (V) | Edges (E) | Faces (F) |
|---|---|---|---|---|---|
| Triangle | 3 | ||||
| Square | 4 | ||||
| Pentagon | 5 | ||||
| Hexagon | 6 | ||||
| Heptagon | 7 | ||||
| Octagon | 8 | ||||
| Nonagon | 9 | ||||
| Decagon | 10 |
⭐ Antiprism Pattern Investigation
Now explore antiprisms! These have triangular faces connecting the bases instead of rectangular ones.
| Build Shape | Base Shape | Sides (n) | Vertices (V) | Edges (E) | Faces (F) |
|---|---|---|---|---|---|
| Triangle | 3 | ||||
| Square | 4 | ||||
| Pentagon | 5 | ||||
| Hexagon | 6 | ||||
| Heptagon | 7 | ||||
| Octagon | 8 | ||||
| Nonagon | 9 | ||||
| Decagon | 10 |
For an antiprism with an n-sided base:
• Vertices = 2n (n vertices on top + n vertices on bottom)
• Edges = 4n (n edges on top + n edges on bottom + 2n diagonal edges)
• Faces = 2n + 2 (2n triangular sides + 2 polygon bases)
• Notice the beautiful pattern: Each column follows a simple formula based on n!
🧮 Pattern Discovery Challenge
After filling in several shapes, try to predict the formulas! Enter your predicted values below:
Vertices =
Edges =
Faces =
Vertices =
Edges =
Faces =