Three-Dimensional Figures - Sixth Grade
Complete Notes & Formulas
1. Three-Dimensional Figures
Definition
A 3D figure is a solid shape that has
• LENGTH (width)
• WIDTH (depth)
• HEIGHT
3D shapes take up SPACE and have VOLUME
Parts of 3D Figures
| Part | Definition | Symbol |
|---|---|---|
| Face | A flat or curved surface | F |
| Edge | Line where two faces meet | E |
| Vertex | Point where edges meet (corner) | V |
2. Polyhedra
What is a Polyhedron?
A polyhedron is a 3D solid made ENTIRELY of
FLAT POLYGONAL FACES
• All faces are flat (no curves)
• All faces are polygons
• All edges are straight lines
Polyhedra vs Non-Polyhedra
| POLYHEDRA ✓ | NOT POLYHEDRA ✗ |
|---|---|
| Cube (all flat faces) | Sphere (curved surface) |
| Rectangular Prism (all flat) | Cylinder (curved surface) |
| Pyramid (all flat faces) | Cone (curved surface) |
| Triangular Prism (all flat) | Hemisphere (curved surface) |
3. Common Three-Dimensional Shapes
| Shape | Description | Faces (F) | Edges (E) | Vertices (V) |
|---|---|---|---|---|
| Cube | 6 square faces, all equal | 6 | 12 | 8 |
| Rectangular Prism | 6 rectangular faces | 6 | 12 | 8 |
| Triangular Prism | 2 triangular, 3 rectangular | 5 | 9 | 6 |
| Square Pyramid | 1 square base, 4 triangular | 5 | 8 | 5 |
| Triangular Pyramid | 4 triangular faces (Tetrahedron) | 4 | 6 | 4 |
| Cylinder | 2 circular bases, 1 curved surface | 3* | 2 | 0 |
| Cone | 1 circular base, 1 curved surface | 2* | 1 | 1 |
| Sphere | 1 curved surface (no faces) | 1* | 0 | 0 |
*Note: Shapes with curved surfaces are NOT polyhedra
4. Euler's Formula for Polyhedra
Euler's Formula
F + V = E + 2
F = Number of Faces
V = Number of Vertices
E = Number of Edges
Example: Verify Euler's Formula for a Cube
Cube has:
F = 6 faces
V = 8 vertices
E = 12 edges
Check Euler's Formula:
F + V = E + 2
6 + 8 = 12 + 2
14 = 14 ✓
It works!
Example: Find Missing Value
Problem: A polyhedron has 8 faces and 12 vertices. Find the number of edges.
F + V = E + 2
8 + 12 = E + 2
20 = E + 2
E = 20 − 2
E = 18
Answer: 18 edges
5. Nets of Three-Dimensional Figures
What is a Net?
A net is a 2D pattern that can be FOLDED
to make a 3D shape
It shows ALL the faces laid out flat
How to Identify a Net
• Count the faces in the net (must match the 3D shape)
• Identify shape of each face (square, rectangle, triangle, etc.)
• Check connections - faces that touch in net will be edges in 3D
• Mentally fold the net to visualize the 3D shape
Common Nets
| 3D Shape | Net Description | Number of Faces |
|---|---|---|
| Cube | 6 connected squares | 6 |
| Rectangular Prism | 6 connected rectangles (some may be squares) | 6 |
| Triangular Prism | 2 triangles, 3 rectangles | 5 |
| Square Pyramid | 1 square, 4 triangles around it | 5 |
| Cylinder | 2 circles, 1 rectangle | 3 |
Important: The same 3D shape can have DIFFERENT nets! A cube has 11 different possible nets.
6. Orthographic Views (Front, Side, Top)
What are Orthographic Views?
Orthographic views show what a 3D object looks like
from different directions (front, side, top)
Each view is a 2D drawing showing only what you see from that angle
The Three Main Views
| View | What You See | Shows |
|---|---|---|
| Front View | Looking at object from the front | Width and Height |
| Side View (Right) | Looking at object from the side | Depth and Height |
| Top View | Looking down at object from above | Width and Depth |
Key Rules
• Only show what's visible from that view
• Hidden parts are not drawn (or shown with dashed lines)
• Each view is 2D (flat drawing)
• Views align - measurements match between views
Example: Rectangular Prism Views
Prism dimensions: 4 units wide × 3 units deep × 2 units tall
Front View: Rectangle (4 wide × 2 tall)
Side View: Rectangle (3 deep × 2 tall)
Top View: Rectangle (4 wide × 3 deep)
7. Describing Three-Dimensional Figures
Key Features to Describe
• Number of faces (and their shapes)
• Number of edges
• Number of vertices
• Base shape (for prisms and pyramids)
• Curved or flat surfaces
• Parallel faces (if any)
Example Descriptions
"A solid with 2 parallel triangular bases and 3 rectangular faces"
Answer: Triangular Prism
"A solid with 1 square base and 4 triangular faces meeting at a point"
Answer: Square Pyramid
"A solid with 6 square faces, all equal in size"
Answer: Cube
Quick Reference: 3D Shapes
| Term | Definition |
|---|---|
| Polyhedron | 3D solid with ALL FLAT faces |
| Face | Flat or curved surface of a 3D shape |
| Edge | Line where two faces meet |
| Vertex | Point where edges meet (corner) |
| Net | 2D pattern that folds into 3D shape |
| Euler's Formula | F + V = E + 2 |
💡 Important Tips to Remember
✓ Polyhedra have ALL FLAT faces (no curves)
✓ Euler's Formula: F + V = E + 2 works for ALL polyhedra
✓ Sphere, cylinder, cone are NOT polyhedra (curved surfaces)
✓ A net shows ALL faces of a 3D shape laid flat
✓ Count faces in net to match 3D shape
✓ Front view shows width and height
✓ Top view shows width and depth
✓ Side view shows depth and height
✓ Prism: 2 parallel bases, rectangular sides
✓ Pyramid: 1 base, triangular sides meeting at apex
🧠 Memory Tricks & Strategies
Polyhedron:
"POLY means MANY, HEDRON means FACES - many flat faces!"
Euler's Formula:
"Faces plus Vertices Equals Edges plus 2 - F + V = E + 2, that's the key!"
Face, Edge, Vertex:
"Faces are FLAT, Edges are LINES, Vertices are POINTS where edges combine!"
Nets:
"A net is flat, but when you FOLD, a 3D shape begins to unfold!"
Prism vs Pyramid:
"Prism has TWO bases that are the same, Pyramid has ONE base - remember the name!"
Orthographic Views:
"Front, side, and top - three views to see, all the parts of 3D geometry!"
Master Three-Dimensional Figures! 📦 🔺 🔵
Remember: Use F + V = E + 2 for all polyhedra!