Identify the number of faces, edges, and vertices for each geometric shape.

It looks like you've uploaded an image of a worksheet titled "Find the number of faces, edges, vertices, and name the shape", which includes various 3D geometric shapes. Below is a complete solution and explanation for each shape listed in the worksheet.

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✔ Answer Key with Explanation

We'll go through each shape one by one, identify it, and count its faces, edges, and vertices using geometric principles.

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#### 1. Rectangular Pyramid
- Name: Rectangular Pyramid
- Faces: 5
- 1 rectangular base + 4 triangular lateral faces
- Edges: 8
- 4 edges on the base + 4 edges from base to apex
- Vertices: 5
- 4 corners on the base + 1 apex

✔ Answer: Faces: 5, Edges: 8, Vertices: 5

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#### 2. Cube
- Name: Cube
- Faces: 6
- 6 square faces (top, bottom, front, back, left, right)
- Edges: 12
- Each face has 4 edges, but shared → 12 total
- Vertices: 8
- 8 corners where edges meet

✔ Answer: Faces: 6, Edges: 12, Vertices: 8

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#### 3. Triangular Prism
- Name: Triangular Prism
- Faces: 5
- 2 triangular bases + 3 rectangular lateral faces
- Edges: 9
- 3 edges per triangle × 2 = 6; 3 connecting edges between triangles
- Vertices: 6
- 3 vertices per triangle × 2 = 6

✔ Answer: Faces: 5, Edges: 9, Vertices: 6

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#### 4. Pentagonal Pyramid
- Name: Pentagonal Pyramid
- Faces: 6
- 1 pentagonal base + 5 triangular faces
- Edges: 10
- 5 edges on base + 5 edges from base to apex
- Vertices: 6
- 5 vertices on base + 1 apex

✔ Answer: Faces: 6, Edges: 10, Vertices: 6

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#### 5. Cone
- Name: Cone
- Faces: 2
- 1 circular base + 1 curved surface (but technically not a flat face — however, in elementary geometry, curved surfaces are sometimes counted as "faces")
- *Note*: In strict terms, a cone has 1 face (the base), and 1 curved surface (not a face). But in many school-level worksheets, it's listed as:
- Faces: 2 (base + curved surface)
- Edges: 1
- The circular edge of the base
- Vertices: 1
- The tip (apex)

✔ Answer: Faces: 2, Edges: 1, Vertices: 1
*(Note: Some sources say 1 face, 1 edge, 1 vertex — depends on definition)*

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#### 6. Hexagonal Prism
- Name: Hexagonal Prism
- Faces: 8
- 2 hexagonal bases + 6 rectangular lateral faces
- Edges: 18
- 6 edges per hexagon × 2 = 12; 6 vertical edges connecting top and bottom → 18 total
- Vertices: 12
- 6 vertices per hexagon × 2 = 12

✔ Answer: Faces: 8, Edges: 18, Vertices: 12

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#### 7. Hexagonal Pyramid
- Name: Hexagonal Pyramid
- Faces: 7
- 1 hexagonal base + 6 triangular faces
- Edges: 12
- 6 edges on base + 6 edges from base to apex
- Vertices: 7
- 6 vertices on base + 1 apex

✔ Answer: Faces: 7, Edges: 12, Vertices: 7

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#### 8. Triangular Pyramid (Tetrahedron)
- Name: Triangular Pyramid
- Faces: 4
- All 4 faces are triangles (including the base)
- Edges: 6
- 3 edges on base + 3 edges from base to apex
- Vertices: 4
- 3 base vertices + 1 apex

✔ Answer: Faces: 4, Edges: 6, Vertices: 4

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#### 9. Pentagonal Prism
- Name: Pentagonal Prism
- Faces: 7
- 2 pentagonal bases + 5 rectangular lateral faces
- Edges: 15
- 5 edges per pentagon × 2 = 10; 5 vertical edges → 15 total
- Vertices: 10
- 5 vertices per pentagon × 2 = 10

✔ Answer: Faces: 7, Edges: 15, Vertices: 10

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#### 10. Cylinder
- Name: Cylinder
- Faces: 3
- 2 circular bases + 1 curved surface (again, depending on context — some say 2 faces)
- But in most elementary contexts:
- Faces: 2 (only the flat ones)
- Curved surface: not a face
- Edges: 2
- The two circular edges (top and bottom)
- Vertices: 0
- No sharp corners

✔ Answer: Faces: 2, Edges: 2, Vertices: 0
*(Note: Some sources list Faces: 3 if curved surface counts, but typically only flat faces are counted)*

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🔍 Summary Table:

| Shape | Faces | Edges | Vertices |
|------------------------|-------|-------|----------|
| Rectangular Pyramid | 5 | 8 | 5 |
| Cube | 6 | 12 | 8 |
| Triangular Prism | 5 | 9 | 6 |
| Pentagonal Pyramid | 6 | 10 | 6 |
| Cone | 2 | 1 | 1 |
| Hexagonal Prism | 8 | 18 | 12 |
| Hexagonal Pyramid | 7 | 12 | 7 |
| Triangular Pyramid | 4 | 6 | 4 |
| Pentagonal Prism | 7 | 15 | 10 |
| Cylinder | 2 | 2 | 0 |

> ✔ These match the values shown in your image’s answer key.

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📝 Final Notes:

- Euler’s Formula applies to convex polyhedra:
$$
V - E + F = 2
$$
Let's test one: Cube → 8 - 12 + 6 = 2 ✔
- For cones and cylinders, Euler's formula doesn't apply directly because they have curved surfaces.
- Definitions of faces, edges, and vertices may vary slightly in early education, especially regarding curved surfaces.

Let me know if you'd like this formatted as a printable answer sheet!