Solid shapes (three-dimensional or 3D shapes) are objects that occupy space. Unlike flat 2D shapes, solids have length, width, and height (depth). Understanding how to visualise and represent them is an important geometric skill.
- Common Solid Shapes:
- Cube: all six faces are equal squares; 8 vertices, 12 edges, 6 faces
- Cuboid (rectangular prism): 6 rectangular faces (opposite faces equal); 8 vertices, 12 edges, 6 faces
- Cylinder: 2 circular bases, 1 curved surface; 2 edges, 0 vertices
- Cone: 1 circular base, 1 curved surface, 1 apex; 1 edge, 1 vertex
- Sphere: perfectly round, no edges, no vertices, 1 curved surface
- Triangular prism: 2 triangular faces, 3 rectangular faces; 6 vertices, 9 edges, 5 faces
- Square pyramid: 1 square base, 4 triangular faces; 5 vertices, 8 edges, 5 faces
Euler's Formula:
For any polyhedron (solid with flat faces): Faces + Vertices - Edges = 2.
F + V - E = 2.
Nets:
A net is a flat 2D pattern that can be folded to form a 3D solid. For example, a cube has multiple possible nets (11 in total), each made from 6 squares arranged in a cross-like or other valid pattern.
- Views of Solid Shapes:
- A solid can be viewed from different directions:
- Front view: what you see from directly in front
- Side view: what you see from the side
- Top view: what you see from directly above (also called plan view)
Drawing and reading these views (also called orthographic projections) helps in understanding the shape of an object without seeing it in 3D.
- Cross-sections:
- A cross-section is the shape you get when a solid is cut by a plane. For example:
- Cutting a cylinder parallel to its base gives a circle.
- Cutting a cube parallel to a face gives a square.
- Cutting a cone through the apex vertically gives a triangle.
How many faces, edges, and vertices does a cube have?
Faces = 6, Edges = 12, Vertices = 8. Check Euler: 6 + 8 - 12 = 2. Correct.
Verify Euler's formula for a triangular prism.
F = 5, V = 6, E = 9. F + V - E = 5 + 6 - 9 = 2. Correct.
What shape is the cross-section of a sphere no matter where you cut it?
Always a circle.
Name the solid with 1 curved surface, 2 edges, and 0 vertices.
A cylinder.
A net has six squares. What solid does it form?
A cube.
What does the top view of a cone look like?
A circle (the base) with a dot in the centre (the apex).
A solid has F = 7, V = 10. Find E using Euler's formula.
F + V - E = 2. 7 + 10 - E = 2. E = 15.
- Key Formulas:
- Euler's formula: F + V - E = 2 (for polyhedra)
Common mistakes
Cylinders and cones are NOT polyhedra (they have curved surfaces), so Euler's formula does not apply to them directly. Do not confuse edges (where two faces meet) with vertices (corners where edges meet).
Summary
3D shapes have faces, edges, and vertices. Euler's formula connects these three counts for any polyhedron. Nets, cross-sections, and different views (front, side, top) help us visualise and understand solids from 2D representations.