Representing 3D in 2D
1. 3D Shapes and Their Parts
Three-dimensional (3D) shapes have three dimensions: LENGTH, BREADTH, and HEIGHT.
Parts of a 3D Shape
| Part | Meaning | Example (Cube) |
|---|---|---|
| Face | Flat surface | 6 faces |
| Edge | Line where two faces meet | 12 edges |
| Vertex (pl. Vertices) | Point where edges meet | 8 vertices |
Common 3D Shapes
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Triangular Prism | 5 | 9 | 6 |
| Square Pyramid | 5 | 8 | 5 |
| Triangular Pyramid (Tetrahedron) | 4 | 6 | 4 |
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
2. Nets of Solids
A NET is a FLAT pattern that can be folded to make a 3D shape.
Cubes: Different Nets
A cube has 11 different nets. Here are a few:
Net 1 (Cross shape):
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Net 2 (T shape):
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Net 3:
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Nets of Common Solids
| Solid | Net Description |
|---|---|
| Cube | 6 connected squares arranged in a cross pattern (11 variations) |
| Cuboid | 3 pairs of rectangles (6 faces total) |
| Cylinder | 2 circles (top and bottom) + 1 rectangle (curved surface) |
| Cone | 1 circle (base) + 1 sector (curved surface) |
| Square Pyramid | 1 square (base) + 4 triangles (lateral faces) |
Worked Example (ICSE 2024, 2 marks)
'Which of the following nets can form a closed cube?' Given 5 nets, identify valid cube nets.
Solution: Valid nets must have exactly 6 squares where each square is connected to at least one other, and when folded, no overlaps and no gaps.
3. Oblique Sketches
An OBLIQUE SKETCH shows a 3D shape where:
- The FRONT face is drawn as it is (true shape).
- The DEPTH is shown using SLANTING lines (usually at 45°).
- Depth measurements are often HALVED.
Example: Cuboid (4 × 3 × 2 units)
Front face (4 × 3) drawn as a rectangle.
Side edges drawn at 45° with half the actual depth (1 unit instead of 2).
Characteristics of Oblique Sketches
- Easier to draw than isometric sketches.
- Front face is UNDISTORTED.
- Depth appears shorter (foreshortened).
4. Isometric Sketches
An ISOMETRIC SKETCH uses ISOMETRIC DOT PAPER to draw 3D shapes.
Rules
- Horizontal lines in real life are drawn at 30° to the horizontal.
- Vertical lines remain vertical.
- Measurements are taken along isometric axes.
- All THREE dimensions are drawn to SCALE.
Drawing a Cube on Isometric Paper
- Draw a vertical line for one edge.
- From the base, draw two lines at 30° (left and right).
- Complete the base parallelogram.
- Draw vertical lines from each vertex of the base.
- Connect the top vertices.
Difference Between Oblique and Isometric
| Feature | Oblique Sketch | Isometric Sketch |
|---|---|---|
| Front face | TRUE shape | Distorted (drawn at 30°) |
| Depth direction | 45° | 30° |
| Measurements | Width and height true, depth half | All three to scale |
| Difficulty | Easier | Moderately difficult |
5. Euler's Formula
Euler's Formula relates the number of faces (F), vertices (V), and edges (E) of a polyhedron:
F + V - E = 2
Verification for a Cube
F = 6, V = 8, E = 12. F + V - E = 6 + 8 - 12 = 14 - 12 = 2. ✓
Using Euler's Formula to Find Missing Values
Example (ICSE 2023, 2 marks): '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 = 18.
Check: Which Solids Satisfy Euler's Formula?
Euler's formula applies to POLYHEDRA (solids with flat faces only). It does NOT apply to cylinders, cones, or spheres.
6. Viewing 3D Shapes from Different Perspectives
- Front view: What you see looking from the front.
- Side view: What you see looking from the side.
- Top view: What you see looking from above.
Different perspectives help us understand the FULL shape of a 3D object.
7. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Identifying nets of solids | 2-3 marks | High |
| Faces, edges, vertices count | 2 marks | Very High |
| Euler's formula | 2-3 marks | High |
| Oblique and isometric sketches | 2-3 marks | Medium |
| Front/Top/Side views | 1-2 marks | Low |
Common Mistakes
- Counting faces that are not flat (curved surfaces — Euler's formula is for polyhedra only).
- Wrong count: edges on a cylinder (2 circular edges — not counted as edges in polyhedra).
- Nets: missing overlaps or gaps when checking if a net forms a solid.
- Isometric sketches: wrong angle (use 30°, not 45°).
Self-Test (5 Questions)
Q1. How many faces does a triangular prism have? (1 mark)
- A) 4
- B) 5
- C) 6
- D) 7
Q2. State Euler's formula. (1 mark)
Q3. 'A polyhedron has 6 faces and 8 edges. Find the number of vertices.' (2 marks)
Q4. 'Which of these is a valid net of a cube: arrangement of 6 squares in a cross?' (1 mark)
Q5. 'A solid has 5 faces, 6 vertices. How many edges does it have? Name the solid.' (3 marks)
Answers
A1. B) 5 (2 triangular + 3 rectangular faces). A2. F + V - E = 2. A3. V = 4. (F + V - E = 2. 6 + V - 8 = 2. V = 4.) A4. YES, a cross-shaped arrangement of 6 squares is a valid cube net. A5. F + V - E = 2. 5 + 6 - E = 2. E = 9. The solid is a square pyramid.
