A relation hiding in the counts
A polygon is a flat shape with straight sides. A prism has two matching polygons as its ends. Counting a prism with a 10-sided base by hand is slow. A pattern does it in seconds.
A pyramid has a polygon as its base. One point sits above the base.
| Solid | Sides of the base | Faces | Corners | Edges |
|---|---|---|---|---|
| Triangular prism | 3 | 5 | 6 | 9 |
| Pentagonal prism | 5 | 7 | 10 | 15 |
| Prism, 10-sided base | 10 | 12 | 20 | 30 |
| Square pyramid | 4 | 5 | 5 | 8 |
| Pentagonal pyramid | 5 | 6 | 6 | 10 |
| Pyramid, 10-sided base | 10 | 11 | 11 | 20 |
Let n be the number of sides of the polygon.
A prism with an n-sided base has n + 2 faces. It has 2n corners and 3n edges. A pyramid with an n-sided base has n + 1 faces. It has n + 1 corners and 2n edges.
Here is why. A prism has 2 ends and n side faces. Each end has n corners and n edges. Another n edges join the ends. A pyramid has n side faces and 1 base. The base has n corners and n edges. The point adds 1 corner. It adds n edges, one to each base corner.
Below, F is faces, C is corners and E is edges.
- A cube: 6 faces, 8 corners, 12 edges.
- F + C − E
- = 6 + 8 − 12
- = 2
Rule: Add the faces and the corners. Take away the edges. Prisms and pyramids give 2.
Watch out!
Take the lid off a cube. The faces go down to 5. The corners and edges stay 8 and 12. Then 5 + 8 − 12 = 1. A box with no lid is not a closed solid.
Add sides. Watch the faces, corners and edges.
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