Properties of 3D Shapes worksheet with illustrations and questions for students to identify and describe geometric solids.

Let's solve this step-by-step. The worksheet asks us to identify 3D shapes and determine their vertices, faces, and edges. Then, in the extension, we need to deduce the names of three unknown shapes (A, B, C) based on their properties.

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🔷 Part 1: Identify the Shapes (1–9)

We’ll go through each shape one by one.

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#### 1) Rectangular Prism (Cuboid)
- Name: Rectangular prism (or cuboid)
- Vertices: 8
- Faces: 6 (all rectangles)
- Edges: 12

✔ *Each corner is a vertex, 6 flat faces, 12 lines where faces meet.*

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#### 2) Triangular Prism
- Name: Triangular prism
- Vertices: 6
- Faces: 5 (2 triangular bases + 3 rectangular sides)
- Edges: 9

✔ Two triangles connected by rectangles.

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#### 3) Cone
- Name: Cone
- Vertices: 1 (the tip)
- Faces: 2 (1 curved surface + 1 circular base)
- Note: Some sources say 1 face (only the base), but technically the curved surface is also a face in 3D geometry.
- However, for elementary levels, often 1 face (base) and 1 curved surface are counted separately — but here we'll follow standard classification:
- Faces: 2 (1 flat, 1 curved)
- Edges: 1 (the circular edge of the base)
- Note: Edges are where two faces meet. The cone has only one edge (the circle).

✔ So:
- Vertices: 1
- Faces: 2
- Edges: 1

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#### 4) Square Pyramid
- Name: Square pyramid
- Vertices: 5 (4 at base, 1 apex)
- Faces: 5 (1 square base + 4 triangular sides)
- Edges: 8 (4 around base, 4 from base to apex)

✔ Classic pyramid with square base.

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#### 5) Cylinder
- Name: Cylinder
- Vertices: 0 (no sharp corners)
- Faces: 3 (2 circular bases + 1 curved surface)
- But some curricula count curved surfaces as not "faces" — but in 3D geometry, they are included.
- So: 3 faces
- Edges: 2 (top and bottom circles)
- Each circular edge counts as an edge.

✔ So:
- Vertices: 0
- Faces: 3
- Edges: 2

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#### 6) Cube
- Name: Cube
- Vertices: 8
- Faces: 6 (all squares)
- Edges: 12

✔ Special case of rectangular prism.

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#### 7) Sphere
- Name: Sphere
- Vertices: 0
- Faces: 0 (smooth surface, no flat faces)
- Edges: 0

✔ A perfectly round 3D shape.

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#### 8) Tetrahedron (Triangular Pyramid)
- Name: Tetrahedron or triangular pyramid
- Vertices: 4
- Faces: 4 (all triangles)
- Edges: 6

✔ Simplest Platonic solid.

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#### 9) Pentagonal Prism
- Name: Pentagonal prism
- Vertices: 10 (5 on top, 5 on bottom)
- Faces: 7 (2 pentagons + 5 rectangles)
- Edges: 15 (5 top, 5 bottom, 5 vertical)

✔ Two pentagons connected by rectangles.

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✔ Summary Table (1–9):

| # | Name | Vertices | Faces | Edges |
|---|------------------------|----------|-------|-------|
| 1 | Rectangular Prism | 8 | 6 | 12 |
| 2 | Triangular Prism | 6 | 5 | 9 |
| 3 | Cone | 1 | 2 | 1 |
| 4 | Square Pyramid | 5 | 5 | 8 |
| 5 | Cylinder | 0 | 3 | 2 |
| 6 | Cube | 8 | 6 | 12 |
| 7 | Sphere | 0 | 0 | 0 |
| 8 | Tetrahedron | 4 | 4 | 6 |
| 9 | Pentagonal Prism | 10 | 7 | 15 |

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🔷 Extension: Identify Shapes A, B, C

We're told:
- A and B are prisms
- C is a pyramid

Let’s use Euler’s formula for polyhedra:
> V – E + F = 2
(Valid for convex polyhedra without holes)

Let’s check if each satisfies this.

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#### A) V=12, F=8, E=18
Check Euler:
12 – 18 + 8 = 2 → ✔ Yes!

Now, since it's a prism, prisms have:
- Two identical polygonal bases
- Number of lateral faces = number of sides of base
- Total faces = base sides + 2
- Vertices = 2 × base vertices
- Edges = 3 × base sides

Let’s suppose base has n sides.

Then:
- Faces: n + 2 = 8 → n = 6 → Hexagon
- Vertices: 2 × 6 = 12 → matches!
- Edges: 3 × 6 = 18 → matches!

✔ So Shape A is a hexagonal prism

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#### B) V=16, F=10, E=24
Check Euler:
16 – 24 + 10 = 2 → ✔ Yes!

Prism again.

Let base have n sides.

- Faces: n + 2 = 10 → n = 8 → Octagon
- Vertices: 2 × 8 = 16 → matches!
- Edges: 3 × 8 = 24 → matches!

✔ So Shape B is an octagonal prism

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#### C) V=6, F=6, E=10
Check Euler:
6 – 10 + 6 = 2 → ✔ Yes!

This is a pyramid.

Pyramids have:
- One base
- Triangular faces meeting at apex
- Faces = base sides + 1
- Vertices = base vertices + 1
- Edges = base sides × 2

Let base have n sides.

Then:
- Faces: n + 1 = 6 → n = 5 → Pentagon
- Vertices: 5 + 1 = 6 → matches!
- Edges: 5 (base) + 5 (from apex to base) = 10 → matches!

✔ So Shape C is a pentagonal pyramid

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✔ Final Answers:

#### Main Section (1–9):

| # | Name | Vertices | Faces | Edges |
|---|------------------------|----------|-------|-------|
| 1 | Rectangular Prism | 8 | 6 | 12 |
| 2 | Triangular Prism | 6 | 5 | 9 |
| 3 | Cone | 1 | 2 | 1 |
| 4 | Square Pyramid | 5 | 5 | 8 |
| 5 | Cylinder | 0 | 3 | 2 |
| 6 | Cube | 8 | 6 | 12 |
| 7 | Sphere | 0 | 0 | 0 |
| 8 | Tetrahedron | 4 | 4 | 6 |
| 9 | Pentagonal Prism | 10 | 7 | 15 |

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#### Extension:

| Shape | Name | Vertices | Faces | Edges |
|-------|--------------------|----------|-------|-------|
| A | Hexagonal Prism | 12 | 8 | 18 |
| B | Octagonal Prism | 16 | 10 | 24 |
| C | Pentagonal Pyramid | 6 | 6 | 10 |

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✔ Explanation Summary:

- Used geometric definitions for each shape.
- Applied Euler’s formula to verify consistency.
- For prisms: used pattern:
- Faces = n + 2
- Vertices = 2n
- Edges = 3n
- For pyramids:
- Faces = n + 1
- Vertices = n + 1
- Edges = 2n

All values match.

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