3D Shapes, Plans, and Elevations
From the phone in your hand to the house you live in, our world is built from 3D shapes. This guide explores their key features — faces, edges, and vertices — and shows you how architects and designers represent them as 2D drawings called plans and elevations.
Common 3D Shapes and Their Properties
Every 3D shape has three key features: Faces (flat surfaces, as opposed to curved surfaces), Edges (where faces meet), and Vertices (corners where edges meet).
Cube
- 6 Faces
- 12 Edges
- 8 Vertices
Cuboid
- 6 Faces
- 12 Edges
- 8 Vertices
Triangular Prism
- 5 Faces
- 9 Edges
- 6 Vertices
Square-Based Pyramid
- 5 Faces
- 8 Edges
- 5 Vertices
Cylinder
- 2 flat Faces + 1 curved surface (which unfurls to a rectangle)
- 2 Edges
- 0 Vertices
Sphere
- 0 flat Faces (1 curved surface)
- 0 Edges
- 0 Vertices
Plans and Elevations
Plans and elevations are 2D drawings of a 3D object from different directions — no perspective, just flat-on views. This is the standard way architects and engineers create blueprints.
The Three Views
- The Plan View is the view from directly above (a bird’s-eye view).
- The Front Elevation is the view from directly in front.
- The Side Elevation is the view from the side (usually the right).
The views must be correctly aligned: the plan sits below the front elevation, and the side elevation sits to the right of the front elevation. Heights and widths must match up across all three views.
Worked Example
Properties of a Pentagonal Prism
How many faces, edges, and vertices does a pentagonal prism have?
- Faces: 2 pentagonal bases + 5 rectangular sides = 7 faces.
- Edges: Each pentagon has 5 edges ($5 \times 2 = 10$) plus 5 edges joining the two bases. Total = 15 edges.
- Vertices: Each pentagon has 5 vertices ($5 \times 2 = 10$). Total = 10 vertices.
Euler’s Formula check: $F + V – E = 7 + 10 – 15 = 2$ ✓
Tutor Insights
🤔 Common Misunderstandings
- Faces vs. curved surfaces: A “face” is a flat surface. A sphere has one continuous curved surface but no faces at all.
- Perspective vs. orthographic: Students often try to draw plans and elevations with perspective (like a photo). Remember — these are flat-on views with no depth or angle.
- What is the base of a prism? A prism may be sitting on one of its rectangular sides, not one of its two identical polygonal bases. The bases are the two identical faces that give the prism its name.
📝 Common Exam Mistakes
- Miscounting faces, edges, or vertices on complex or unusual shapes.
- Incorrect alignment of plans and elevations — the side elevation must be to the right of the front, and heights must match.
- Missing hidden lines in a view — dashed lines must be used for edges you cannot see from that direction.
- Not using a ruler, leading to inaccurate and unmarked drawings.
Practice Questions
- A standard six-sided dice is a cube. How many faces, edges, and vertices does it have?
- A party hat is a cone. How many flat faces, edges, and vertices does it have?
- A Toblerone box is a triangular prism. How many faces, edges, and vertices does it have?
- A hexagonal prism has a regular hexagon as its cross-section. How many faces, edges, and vertices does it have? Verify using Euler’s formula.
Show Answers
- 6 faces, 12 edges, 8 vertices.
- 1 flat face (the circular base), 1 curved surface, 1 edge (the circular rim), 1 vertex (the apex).
- 5 faces (2 triangular + 3 rectangular), 9 edges, 6 vertices.
- 8 faces (2 hexagonal + 6 rectangular), 18 edges, 12 vertices. Euler check: $F + V – E = 8 + 12 – 18 = 2$ ✓
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