Three-Dimensional Figures - Grade 8
1. Parts of Three-Dimensional Figures
Three-dimensional (3D) figures have three main components: faces, edges, and vertices.
Key Terms:
1. Face:
- A flat or curved surface of a 3D shape
- The 2D shapes that make up the solid
- Example: A cube has 6 square faces
2. Edge:
- A line segment where two faces meet
- The boundary between two surfaces
- Example: A cube has 12 edges
3. Vertex (plural: Vertices):
- A corner point where three or more edges meet
- The sharp point of a 3D shape
- Example: A cube has 8 vertices
Common 3D Shapes:
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Rectangular Prism (Cuboid) | 6 | 12 | 8 |
| Triangular Prism | 5 | 9 | 6 |
| Square Pyramid | 5 | 8 | 5 |
| Triangular Pyramid (Tetrahedron) | 4 | 6 | 4 |
| Hexagonal Prism | 8 | 18 | 12 |
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
Euler's Formula (for Polyhedra):
\( F + V - E = 2 \)
where \( F \) = number of faces, \( V \) = number of vertices, \( E \) = number of edges
Examples Using Euler's Formula:
Example 1: Verify Euler's formula for a cube.
Cube: \( F = 6 \), \( V = 8 \), \( E = 12 \)
\( F + V - E = 6 + 8 - 12 = 2 \) ✓
Example 2: A polyhedron has 8 faces and 6 vertices. Find the number of edges.
\( F + V - E = 2 \)
\( 8 + 6 - E = 2 \)
\( 14 - E = 2 \) → \( E = 12 \) edges
2. Nets of Three-Dimensional Figures
Definition: A net is a two-dimensional pattern that can be folded to form a three-dimensional figure.
Key Concepts:
- A net shows all faces of a 3D shape laid out flat
- When folded along the edges, the net forms the 3D shape
- One 3D shape can have multiple different nets
- Nets help visualize surface area of 3D shapes
Common Nets:
1. Cube Net:
- Consists of 6 connected squares
- A cube has 11 different possible nets
- Most common: cross-shaped pattern
2. Rectangular Prism (Cuboid) Net:
- Consists of 6 rectangles
- Opposite faces are congruent
- Contains 3 pairs of identical rectangles
3. Triangular Prism Net:
- 2 triangular bases (at the ends)
- 3 rectangular lateral faces
- Total: 5 faces
4. Square Pyramid Net:
- 1 square base
- 4 triangular lateral faces
- Triangles meet at apex when folded
5. Cylinder Net:
- 2 circles (top and bottom)
- 1 rectangle (curved surface when rolled)
- Rectangle width = circumference of circle
6. Cone Net:
- 1 circle (base)
- 1 sector (curved surface)
- Sector forms cone when wrapped
Using Nets to Find Surface Area:
Steps:
- Draw or identify the net of the 3D shape
- Calculate the area of each face in the net
- Add all the areas together
- The sum is the surface area of the 3D shape
Example:
Find the surface area of a cube with side length 5 cm using a net.
Net consists of 6 squares, each with area \( 5 \times 5 = 25 \) cm²
Total surface area = \( 6 \times 25 = 150 \) cm²
Tips for Identifying Nets:
- Count the number of faces in the net
- Identify the shapes of each face
- Look for base(s) and lateral faces
- Imagine folding the net mentally
3. Front, Side, and Top View
Definition: Orthogonal views are 2D representations of 3D objects as seen from different directions.
Three Standard Views:
1. Front View (Elevation):
- What you see when looking directly at the front of the object
- Shows height and width
- Faces parallel to the front are shown in true size
2. Side View (Profile/Side Elevation):
- What you see when looking from the side (usually right side)
- Shows height and depth
- Perpendicular to the front view
3. Top View (Plan):
- What you see when looking down from directly above
- Shows width and depth
- Perpendicular to both front and side views
Common Examples:
| 3D Shape | Front View | Side View | Top View |
|---|---|---|---|
| Cube | Square | Square | Square |
| Rectangular Prism | Rectangle | Rectangle | Rectangle |
| Cylinder | Rectangle | Rectangle | Circle |
| Cone | Triangle | Triangle | Circle |
| Sphere | Circle | Circle | Circle |
| Square Pyramid | Triangle | Triangle | Square |
Key Points:
- Views show the outline/silhouette of the shape from that direction
- Hidden edges are usually not shown in orthogonal views
- Three views together can uniquely identify a 3D shape
- Views help in technical drawing and engineering
Practice Tips:
- Imagine rotating the object in your mind
- Focus on what edges are visible from each direction
- Draw simple sketches to visualize better
- Use physical models if available
4. Similar Solids
Definition: Two solids are similar if they have the same shape but not necessarily the same size.
Properties of Similar Solids:
- All corresponding angles are equal
- All corresponding linear dimensions are proportional
- Same shape, different size
- Related by a scale factor
Scale Factor (k):
\( k = \frac{\text{Linear dimension of Figure 2}}{\text{Linear dimension of Figure 1}} \)
The scale factor is the ratio of corresponding lengths
Key Formulas for Similar Solids:
1. Linear Dimension Ratio: \( \frac{l_1}{l_2} = k \)
2. Surface Area Ratio: \( \frac{SA_1}{SA_2} = k^2 \)
The ratio of surface areas equals the square of the scale factor
3. Volume Ratio: \( \frac{V_1}{V_2} = k^3 \)
The ratio of volumes equals the cube of the scale factor
Summary Table:
| Measure | Ratio | Power of k |
|---|---|---|
| Length, Width, Height | \( k : 1 \) | \( k^1 \) |
| Perimeter | \( k : 1 \) | \( k^1 \) |
| Surface Area | \( k^2 : 1 \) | \( k^2 \) |
| Volume | \( k^3 : 1 \) | \( k^3 \) |
Examples:
Example 1: Two similar cubes have edge lengths 4 cm and 8 cm. Find the ratio of their surface areas and volumes.
Scale factor: \( k = \frac{8}{4} = 2 \)
Surface area ratio: \( k^2 = 2^2 = 4 \) or \( 4:1 \)
Volume ratio: \( k^3 = 2^3 = 8 \) or \( 8:1 \)
Example 2: Two similar cylinders have heights 5 cm and 15 cm. If the smaller cylinder has surface area 100 cm², find the surface area of the larger cylinder.
Scale factor: \( k = \frac{15}{5} = 3 \)
Surface area ratio: \( k^2 = 3^2 = 9 \)
\( \frac{SA_{\text{large}}}{SA_{\text{small}}} = 9 \)
\( \frac{SA_{\text{large}}}{100} = 9 \)
\( SA_{\text{large}} = 900 \) cm²
Example 3: Two similar rectangular prisms have volumes 64 cm³ and 216 cm³. If the smaller prism has length 4 cm, find the length of the larger prism.
Volume ratio: \( \frac{V_1}{V_2} = \frac{64}{216} = \frac{8}{27} = \frac{2^3}{3^3} \)
Scale factor: \( k = \frac{3}{2} = 1.5 \)
\( \frac{l_{\text{large}}}{l_{\text{small}}} = 1.5 \)
\( \frac{l_{\text{large}}}{4} = 1.5 \)
\( l_{\text{large}} = 6 \) cm
Example 4: Two similar cones have volumes in the ratio 8:27. What is the ratio of their heights?
Volume ratio = \( k^3 = \frac{8}{27} = \frac{2^3}{3^3} \)
Scale factor: \( k = \frac{2}{3} \)
Height ratio: \( 2:3 \)
5. Volume and Surface Area Formulas
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | \( V = s^3 \) | \( SA = 6s^2 \) |
| Rectangular Prism | \( V = lwh \) | \( SA = 2(lw + lh + wh) \) |
| Triangular Prism | \( V = \frac{1}{2}bhl \) | \( SA = bh + (a+b+c)l \) |
| Cylinder | \( V = \pi r^2 h \) | \( SA = 2\pi r^2 + 2\pi rh \) |
| Cone | \( V = \frac{1}{3}\pi r^2 h \) | \( SA = \pi r^2 + \pi rl \) |
| Sphere | \( V = \frac{4}{3}\pi r^3 \) | \( SA = 4\pi r^2 \) |
| Pyramid | \( V = \frac{1}{3}Bh \) | \( SA = B + \frac{1}{2}Pl \) |
Key: \( s \) = side, \( l \) = length, \( w \) = width, \( h \) = height, \( r \) = radius, \( B \) = base area, \( P \) = base perimeter, \( l \) = slant height
Quick Reference: 3D Figures
Euler's Formula:
\( F + V - E = 2 \)
Similar Solids Ratios:
- Linear dimensions: \( k : 1 \)
- Surface area: \( k^2 : 1 \)
- Volume: \( k^3 : 1 \)
Remember:
- A net is a 2D pattern that folds into a 3D shape
- Orthogonal views show front, side, and top perspectives
- Similar solids have the same shape but different sizes
💡 Key Tips for Three-Dimensional Figures
- ✓ Face = flat surface, Edge = where faces meet, Vertex = corner point
- ✓ Euler's Formula works for polyhedra: F + V - E = 2
- ✓ Nets help calculate surface area by adding all face areas
- ✓ One 3D shape can have multiple different nets
- ✓ Three orthogonal views uniquely identify a 3D shape
- ✓ Visualize: imagine rotating the object in your mind
- ✓ For similar solids: Linear ratio = k, Area ratio = k², Volume ratio = k³
- ✓ Scale factor is always: larger ÷ smaller
- ✓ If volume ratio is known, take cube root to find scale factor
- ✓ If area ratio is known, take square root to find scale factor
- ✓ Draw diagrams to help visualize 3D shapes
- ✓ Practice with physical models when possible