Answer key for 3D shape properties worksheet with diagrams and numerical data.

This image is an answer sheet for a 3D shape properties worksheet, titled "3D Shape Properties Sheet 5 Answers", from the website Math-Salamanders.com. It provides the number of faces (F), edges (E), and vertices (V) for various 3D shapes.

Let’s go through each shape and explain the answers, including why some values have two possible answers (e.g., "1 or 2") — this arises due to different definitions in geometry regarding curved surfaces and edges.

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🔷 1. Cube

- F: 6 – 6 flat square faces
- E: 12 – 12 straight edges where faces meet
- V: 8 – 8 corners (vertices)
✔ All are flat, so no ambiguity.

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🔷 2. Triangular Prism

- F: 5 – 2 triangular bases + 3 rectangular sides = 5 faces
- E: 9 – Each triangle has 3 edges; shared edges between faces → total 9
- V: 6 – 3 vertices per triangle, but shared → 6 total
✔ Standard polyhedron with all flat faces.

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🔷 3. Triangular Pyramid (Tetrahedron)

- F: 4 – 4 triangular faces
- E: 6 – Each edge shared by two faces → 6 edges
- V: 4 – 4 corners
✔ A regular pyramid with all flat faces.

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🔷 4. Cuboid (Rectangular Prism)

- F: 6 – 6 rectangular faces
- E: 12 – Like cube, but rectangles instead of squares
- V: 8 – Same as cube
✔ Flat faces only.

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🔷 5. Cone

Here's where things get ambiguous — depending on how you define face, edge, and vertex.

#### ✔ Two interpretations:
| Interpretation | Faces (F) | Edges (E) | Vertices (V) |
|----------------|-----------|-----------|--------------|
| Strict (flat only) | 1 (base only) | 0 (no straight edges) | 0 (no vertex at tip) |
| Loose (includes curved surface) | 2 (base + curved surface) | 1 (circular edge of base) | 1 (apex/vertex) |

So:
- F: 1 or 2
- E: 0 or 1
- V: 0 or 1

> ⚠️ Note: The curved surface is not considered a face in strict Euclidean geometry (faces must be flat), but sometimes it's counted in basic math education.

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🔷 6. Cylinder

Again, depends on interpretation.

| Interpretation | Faces (F) | Edges (E) | Vertices (V) |
|----------------|-----------|-----------|--------------|
| Strict | 2 (top and bottom circles) | 0 (no straight edges) | 0 (no vertices) |
| Loose | 3 (2 flat + 1 curved side) | 2 (top and bottom circular edges) | 0 (still no sharp points) |

So:
- F: 2 or 3
- E: 0 or 2
- V: 0

> 🔍 No vertices because there are no sharp corners — even though there are circular edges, they aren't vertices.

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🔷 7. Sphere

- F: 0 or 1 – If you count the whole surface as one face (curved), or none (only if flat faces count)
- E: 0 – No edges
- V: 0 – No vertices

> 🌐 A sphere has no flat faces, no edges, and no vertices in standard geometry.

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🔷 8. Square Pyramid

- F: 5 – 1 square base + 4 triangular sides
- E: 8 – 4 edges on base + 4 slant edges to apex = 8
- V: 5 – 4 base corners + 1 apex

✔ Clear-cut, all flat faces.

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✔ Summary Table:

| Shape | F | E | V |
|-------------------|-------|-------|-------|
| Cube | 6 | 12 | 8 |
| Triangular Prism | 5 | 9 | 6 |
| Triangular Pyramid| 4 | 6 | 4 |
| Cuboid | 6 | 12 | 8 |
| Cone | 1 or 2| 0 or 1| 0 or 1|
| Cylinder | 2 or 3| 0 or 2| 0 |
| Sphere | 0 or 1| 0 | 0 |
| Square Pyramid | 5 | 8 | 5 |

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📝 Key Takeaway:

The note at the top is crucial:
> *"the answer to some of these questions depends on whether or not you count a curved surface as a face, and whether or not you can have a curved edge."*

In elementary math, especially in primary school, students often learn that:
- A face is a flat surface.
- An edge is a straight line where two faces meet.
- A vertex is a corner point.

But sometimes, curved surfaces are treated as "faces" in simplified models, leading to multiple acceptable answers.

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💡 Final Tip:

When teaching or learning:
- Use strict definitions for accuracy in higher math.
- Accept flexible answers when working with young learners or in basic classification tasks.

Let me know if you'd like a printable version or a student-friendly explanation!