Representing 3D in 2D
1. 3D Shapes — Solids
A 3D (three-dimensional) shape has LENGTH, BREADTH, and HEIGHT.
| Solid | Faces | Edges | Vertices | Example |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | Dice |
| Cuboid | 6 | 12 | 8 | Brick |
| Triangular Prism | 5 | 9 | 6 | Toblerone box |
| Square Pyramid | 5 | 8 | 5 | Egyptian pyramid |
| Cylinder | 3 | 2 | 0 | Can |
| Cone | 2 | 1 | 1 | Ice cream cone |
| Sphere | 1 | 0 | 0 | Ball |
'Faces are the FLAT surfaces. Edges are the LINE segments where faces meet. Vertices are the CORNERS where edges meet.'
2. Nets of Solids
A net is a FLAT, TWO-dimensional shape that can be FOLDED to form a THREE-dimensional solid.
Cube — 11 Different Nets
A cube has 11 distinct nets. In each net, 6 squares are arranged so that when folded, they form a cube.
Properties of a cube net:
- Must have EXACTLY 6 squares
- Squares must connect along FULL edges (not at corners)
- No overlapping when folded
Cuboid Net
A cuboid net has 6 rectangles with matching dimensions.
Worked Example: Draw the net of a cuboid with dimensions 4 cm × 3 cm × 2 cm.
The net consists of:
- Bottom face: 4 × 3
- Top face: 4 × 3
- Front and back: 4 × 2 each
- Left and right sides: 3 × 2 each
Other Nets
- Cylinder net: TWO circles (top and bottom) + ONE rectangle (curved surface)
- Cone net: ONE sector of a circle (curved surface) + ONE circle (base)
- Square pyramid net: ONE square (base) + FOUR triangles (lateral faces)
3. Euler's Formula
F + V — E = 2
Where F = number of FACES, V = number of VERTICES, E = number of EDGES.
'Euler's formula is TRUE for ALL convex polyhedra. It is a FUNDAMENTAL relationship in geometry.'
| Solid | F | V | E | F + V — E |
|---|---|---|---|---|
| Cube | 6 | 8 | 12 | 6+8-12 = 2 |
| Cuboid | 6 | 8 | 12 | 6+8-12 = 2 |
| Triangular Prism | 5 | 6 | 9 | 5+6-9 = 2 |
| Square Pyramid | 5 | 5 | 8 | 5+5-8 = 2 |
Worked Example: 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
Worked Example: A polyhedron has 6 faces and 12 edges. Find the number of vertices.
6 + V — 12 = 2 V — 6 = 2 V = 8
4. Isometric Sketches
An isometric sketch shows a 3D shape on isometric dot paper where:
- Three axes are drawn at 120° to each other
- Measurements along these axes are to SCALE
- Depth is VISUALLY represented
'Isometric drawings give a REALISTIC 3D feel. Unlike oblique sketches, all three dimensions are drawn to scale.'
Rules for Isometric Drawing
- Draw three axes: vertical, left 30° to horizontal, right 30° to horizontal (120° apart).
- Measure all distances along these axes (not along horizontals).
- Hidden edges are usually shown with DASHED lines.
5. Oblique Sketches
An oblique sketch is a simpler way to show 3D shapes:
- The front face is drawn TRUE TO SHAPE
- Depth is shown at a 45° angle (usually half scale)
'Oblique sketches are EASIER to draw than isometric sketches. The front face looks the same as the actual 2D shape.'
Comparison:
| Aspect | Isometric | Oblique |
|---|---|---|
| Front face | Distorted | True shape |
| Ease of drawing | Harder | Easier |
| Depth accuracy | To scale | Often half scale |
| Realism | More realistic | Less realistic |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Counting a face that is not a polygon' | Euler's formula applies to POLYHEDRA (faces are polygons). Cylinders and cones are NOT polyhedra |
| 'Squares touching only at corners in a net' | Squares in a net must share a FULL edge, not just a point |
| 'Drawing isometric horizontal lines' | In isometric drawing, there are NO true horizontals except the axes |
| 'Forgetting to count hidden faces' | Include ALL faces — visible and hidden — in the count |
ICSE Exam Focus (4–6 marks)
- 2-mark questions: Identify faces, edges, vertices of a solid
- 3-mark questions: Apply Euler's formula to find missing values
- 4-mark questions: Draw nets of cubes, cuboids, or cylinders
- 6-mark questions: Verify Euler's formula for a given solid
Self-Test
Q1. How many faces, edges, and vertices does a cube have? A1. F = 6, V = 8, E = 12. Euler's formula: 6 + 8 — 12 = 2 ✓.
Q2. A polyhedron has 7 faces and 10 vertices. Find the number of edges. A2. F + V — E = 2 → 7 + 10 — E = 2 → E = 15.
Q3. Can a polyhedron have 4 faces, 4 vertices, and 6 edges? A3. F+V—E = 4+4—6 = 2 ✓. Yes, this is a TETRAHEDRON (triangular pyramid).
Q4. Draw the net of a cylinder and label its parts. A4. A cylinder net has: (1) Two circles for the top and bottom bases. (2) One rectangle (width = circumference of circle = 2πr, height = height of cylinder).
Q5. What is the difference between an isometric and an oblique sketch? A5. In isometric sketches, all three axes are at 120° and scaled equally. In oblique sketches, the front face is true-to-shape and depth is at 45° (often half scale). Isometric is more realistic but harder to draw.
Q6. A hexagonal prism has 8 faces and 18 edges. How many vertices does it have? A6. F + V — E = 2 → 8 + V — 18 = 2 → V = 12.
