Properties of 3D Shapes | PDF printable Geometry Worksheets - Free Printable

Let's solve this step-by-step. The worksheet asks us to identify 3D shapes and determine their properties: name, vertices, faces, and edges.

We'll go through each shape one by one and then tackle the extension questions.

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🔷 Basic Definitions

- Vertices: Corners where edges meet.
- Faces: Flat surfaces (or curved in some cases).
- Edges: Lines where two faces meet.

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## ✔ Part 1: Identify Shapes 1–9

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1) Rectangular Prism (Cuboid)

- Name: Rectangular prism
- Vertices: 8 (corners)
- Faces: 6 (all rectangles)
- Edges: 12

✔ *Answer:*
Name: Rectangular prism
Vertices: 8
Faces: 6
Edges: 12

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2) Triangular Prism

- Two triangular bases and three rectangular sides.
- Name: Triangular prism
- Vertices: 6 (3 on each triangle)
- Faces: 5 (2 triangles + 3 rectangles)
- Edges: 9

✔ *Answer:*
Name: Triangular prism
Vertices: 6
Faces: 5
Edges: 9

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3) Cone

- One circular base and a curved surface meeting at a point.
- Name: Cone
- Vertices: 1 (the tip or apex)
- Faces: 2 (1 flat circular face + 1 curved surface — but sometimes curved is not counted as a face; in elementary math, often counted as 1 face)
- However, in most primary-level curricula, the cone has:
- 1 face (the base)
- 1 curved surface (not always counted as a face)
- But for consistency with standard classification:
- Faces: 2 (base + curved surface — though curved isn't flat)
- But often only flat faces are counted → so 1 face
- Let’s follow common practice in primary education:
- Faces: 1 (only the base)
- Curved surface is not considered a "face"
- Edges: 0 (no straight edges — just a curve)
- Vertex: 1 (the tip)

Wait — let's clarify:

In elementary geometry, for cones:
- Vertex: 1 (tip)
- Faces: 1 (base)
- Edges: 0 (no straight edges — the circle is curved)
- Curved surface is not a face in many systems

But sometimes curved surfaces are included as "faces" — however, faces usually mean flat surfaces.

So:
- Faces: 1 (the base)
- Edges: 0 (no line segments)
- Vertices: 1 (apex)

✔ *Answer:*
Name: Cone
Vertices: 1
Faces: 1
Edges: 0

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4) Square Pyramid

- A square base with 4 triangular faces meeting at a point.
- Name: Square pyramid
- Vertices: 5 (4 base corners + 1 apex)
- Faces: 5 (1 square base + 4 triangles)
- Edges: 8 (4 base edges + 4 from base to apex)

✔ *Answer:*
Name: Square pyramid
Vertices: 5
Faces: 5
Edges: 8

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5) Cylinder

- Two circular bases and one curved surface.
- Name: Cylinder
- Vertices: 0 (no corners)
- Faces: 2 (top and bottom circles)
- Edges: 0 (curved side doesn’t have edges)
- Note: Some say there are 2 edges (the circular rims), but in standard definitions, edges are straight lines, so cylinders have 0 edges

✔ *Answer:*
Name: Cylinder
Vertices: 0
Faces: 2
Edges: 0

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6) Cube

- All sides equal — special case of rectangular prism.
- Name: Cube
- Vertices: 8
- Faces: 6
- Edges: 12

✔ *Answer:*
Name: Cube
Vertices: 8
Faces: 6
Edges: 12

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7) Sphere

- Completely round shape.
- Name: Sphere
- Vertices: 0
- Faces: 0 (no flat faces)
- Edges: 0

✔ *Answer:*
Name: Sphere
Vertices: 0
Faces: 0
Edges: 0

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8) Tetrahedron (Triangular Pyramid)

- 4 triangular faces, all equilateral or similar.
- Name: Tetrahedron (or triangular pyramid)
- Vertices: 4
- Faces: 4
- Edges: 6

✔ *Answer:*
Name: Tetrahedron
Vertices: 4
Faces: 4
Edges: 6

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9) Pentagonal Prism

- Two pentagonal bases and five rectangular sides.
- Name: Pentagonal prism
- Vertices: 10 (5 per base)
- Faces: 7 (2 pentagons + 5 rectangles)
- Edges: 15

✔ *Answer:*
Name: Pentagonal prism
Vertices: 10
Faces: 7
Edges: 15

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## ✔ Extension: Identify Shapes A, B, C

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

Use Euler’s formula:
For convex polyhedra:
> V – E + F = 2

Let’s check if it holds.

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A)

- Vertices: 12
- Faces: 8
- Edges: 18

Check Euler:
12 – 18 + 8 = 2 → ✔️ Valid

Now, since it's a prism, and prisms have:
- 2 identical bases
- Number of lateral faces = number of sides in base
- So total faces = 2 + n
- Vertices = 2n
- Edges = 3n

Here:
- Faces = 8 → 2 + n = 8 → n = 6 → hexagon
- Vertices = 2n = 12 → matches
- Edges = 3n = 18 → matches

✔ So, Hexagonal prism

Answer:
Name: Hexagonal prism

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B)

- Vertices: 16
- Faces: 10
- Edges: 24

Check Euler:
16 – 24 + 10 = 2 → ✔️ Valid

Prism:
- Faces = 2 + n → 10 = 2 + n → n = 8 → octagon
- Vertices = 2n = 16 → matches
- Edges = 3n = 24 → matches

✔ So, Octagonal prism

Answer:
Name: Octagonal prism

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C)

- Vertices: 6
- Faces: 6
- Edges: 10

Euler: 6 – 10 + 6 = 2 → ✔️ Valid

It's a pyramid, so:
- One base, rest are triangular faces meeting at apex
- Number of triangular faces = number of sides in base
- Total faces = 1 (base) + n (triangles) = n+1
- Here, faces = 6 → n+1 = 6 → n = 5 → pentagon
- So base is pentagon
- Vertices: n (base) + 1 (apex) = 6 → matches
- Edges: n (base) + n (from base to apex) = 2n = 10 → n=5 → matches

✔ So, Pentagonal pyramid

Answer:
Name: Pentagonal pyramid

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## ✔ Final Answers Summary

| Shape | Name | Vertices | Faces | Edges |
|-------|------------------------|----------|-------|-------|
| 1 | Rectangular prism | 8 | 6 | 12 |
| 2 | Triangular prism | 6 | 5 | 9 |
| 3 | Cone | 1 | 1 | 0 |
| 4 | Square pyramid | 5 | 5 | 8 |
| 5 | Cylinder | 0 | 2 | 0 |
| 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 Answers

- A) Hexagonal prism
- B) Octagonal prism
- C) Pentagonal pyramid

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✔ Final Answer Key (for student):

Main Section:

1. Rectangular prism: V=8, F=6, E=12
2. Triangular prism: V=6, F=5, E=9
3. Cone: V=1, F=1, E=0
4. Square pyramid: V=5, F=5, E=8
5. Cylinder: V=0, F=2, E=0
6. Cube: V=8, F=6, E=12
7. Sphere: V=0, F=0, E=0
8. Tetrahedron: V=4, F=4, E=6
9. Pentagonal prism: V=10, F=7, E=15

Extension:

- A) Hexagonal prism
- B) Octagonal prism
- C) Pentagonal pyramid

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