3D Shapes
3D shapes have length, width, and height. We classify them by their faces, edges, and vertices and explore their nets, cross-sections, and properties.
Key formula: Euler's Formula โ for any polyhedron: F + V โ E = 2
Click a shape above to see its properties.
๐ Learn
Parts of a 3D Shape
- Face: a flat or curved surface
- Edge: where two faces meet
- Vertex (pl. Vertices): a corner where three or more edges meet
Euler's Formula (polyhedra only): F + V โ E = 2
Example: Cube โ 6 + 8 โ 12 = 2 โ
๐ก Cylinders, cones, and spheres are NOT polyhedra (they have curved surfaces), so Euler's formula doesn't apply in the usual form.
Polyhedra โ Prisms and Pyramids
| Shape | Faces (F) | Vertices (V) | Edges (E) |
|---|
| Cube | 6 | 8 | 12 |
| Cuboid | 6 | 8 | 12 |
| Triangular prism | 5 | 6 | 9 |
| Pentagonal prism | 7 | 10 | 15 |
| Triangular pyramid (tetrahedron) | 4 | 4 | 6 |
| Square pyramid | 5 | 5 | 8 |
| Pentagonal pyramid | 6 | 6 | 10 |
Prism rule: n-sided base โ F = n+2, V = 2n, E = 3n
Pyramid rule: n-sided base โ F = n+1, V = n+1, E = 2n
Nets
A net is the flat shape you get when you unfold a 3D solid along its edges. The net contains all the faces joined together.
- Cube net: 6 squares arranged in a cross (or T-shape etc. โ 11 valid nets)
- Cuboid net: 3 pairs of rectangles
- Cylinder net: 2 circles + 1 rectangle
- Triangular prism net: 2 triangles + 3 rectangles
- Square pyramid net: 1 square + 4 triangles
- Cone net: 1 circle + 1 sector (fan shape)
๐ก When drawing a net, make sure every face is present and no face overlaps another.
Cross-Sections
A cross-section is the 2D shape you get when you cut a 3D solid with a flat plane.
- Cube: horizontal cut โ square; diagonal cut โ rectangle; corner cut โ triangle or hexagon
- Cylinder: horizontal โ circle; vertical โ rectangle; angled โ ellipse
- Cone: horizontal โ circle; through tip โ triangle; angled โ parabola/ellipse
- Sphere: any cut through centre โ circle with r = sphere radius; off-centre โ smaller circle
- Prism: parallel to base โ same shape as base
- Pyramid: parallel to base โ similar (scaled) version of base
Plans and Elevations
A 3D shape can be represented by three 2D views:
- Plan (top view): looking down from above
- Front elevation: looking from the front
- Side elevation: looking from the side
๐ก Hidden edges are shown as dashed lines. Use dotted lines for edges you cannot see from that direction.
โ๏ธ Worked Examples
Example 1: Verifying Euler's Formula
Show that a triangular prism satisfies Euler's formula.
Faces: 2 triangles + 3 rectangles = 5 faces (F = 5)
Vertices: 2 triangles ร 3 corners = 6 vertices (V = 6)
Edges: 3 on each triangle + 3 connecting = 9 edges (E = 9)
F + V โ E = 5 + 6 โ 9 = 2 โ
Example 2: Finding missing edges using Euler's Formula
A polyhedron has 8 faces and 12 vertices. How many edges does it have?
F + V โ E = 2
8 + 12 โ E = 2
20 โ E = 2
E = 18
Example 3: Identifying cross-sections
A cylinder is cut by a horizontal plane and then by a vertical plane through its axis. Describe each cross-section.
Horizontal cut โ circle (same as the base)
Vertical cut through axis โ rectangle (width = diameter, height = cylinder height)
Example 4: Prisms and pyramids formula
A hexagonal prism has a hexagonal base. Find F, V, E.
n = 6 (hexagon)
F = n + 2 = 8, V = 2n = 12, E = 3n = 18
Check: F + V โ E = 8 + 12 โ 18 = 2 โ
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