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Class 8 Exploring Some Geometric Themes: cheat sheet
CBSE Mathematics. Faces, vertices and edges of solids and Euler’s formula, prisms and pyramids, nets, painted cubes, and fractals that repeat a pattern at smaller sizes.
Key ideas
- For any polyhedron without holes, V − E + F = 2.
- A net is a flat shape that folds into a solid; a cube has 11 different nets.
- In a painted cube cut into small cubes, corner cubes have 3 painted faces, edge cubes 2, face cubes 1, and inside cubes none.
- Sierpinski triangle: each step removes the middle triangle from every triangle, leaving 3 times as many triangles.
- Koch snowflake: each step replaces every side with 4 sides, each a third as long, so the perimeter grows.
Remember
| Euler’s formula | V − E + F = 2 |
| Prism on an n-sided base | V = 2n, E = 3n, F = n + 2 |
| Pyramid on an n-sided base | V = n + 1, E = 2n, F = n + 1 |
| Painted n × n × n cube | 3 faces: 8; 2: 12(n − 2); 1: 6(n − 2)²; 0: (n − 2)³ |
| Fractal steps | Sierpinski: triangles × 3; Koch: perimeter × 4/3 |
Watch out
- Counting an edge twice when it is shared by two faces.
- Forgetting the 8 corner cubes when counting painted cubes.
- Mixing up the prism and pyramid counts.
Worked example
A 8 × 5 × 5 block is painted on the outside and cut into 200 unit cubes. How many cubes have paint on exactly 3, 2, 1 and 0 faces?
Working: corners 8; edges 4 × (6 + 3 + 3); faces 2 × (18 + 9 + 18); inside 6 × 3 × 3
Answer: 3 faces: 8; 2 faces: 48; 1 face: 90; none: 54
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