Representing 3-d in 2-d
179. Faces, Edges, and Vertices · Counting the parts of a polyhedron
Euler's formula F + V - E = 2 holds for the cube.
Euler's Formula for any closed convex polyhedron: F + V − E = 2, where F = faces, V = vertices, E = edges. Cube: 6 + 8 − 12 = 2 ✓. Tetrahedron: 4 + 4 − 6 = 2 ✓. A beautiful invariant of 3D shapes.
What this lesson covers
Try to break it
Drag vertex A. The edges connected to A move with it, and the faces sharing A reshape together. Notice that every edge is the meeting of two faces, and every vertex is where three or more edges meet. Try to drag A so an edge breaks off from its faces — impossible.
How you build it
Construct a cuboid.
- Draw the front face of the cuboid using the square tool.
- Draw the back face using the square tool — place it slightly up and to the right of the front face to give the cuboid depth.
- Connect the top-left corners: draw a segment from the top-left corner of the front face to the top-left corner of the back face.
- Connect the top-right corners: draw a segment from the top-right corner of the front face to the top-right corner of the back face.
- Connect the bottom-right corners: draw a segment from the bottom-right corner of the front face to the bottom-right corner of the back face.
- Connect the bottom-left corners: draw a segment from the bottom-left corner of the front face to the bottom-left corner of the back face. The cuboid is now complete — 6 faces, 12 edges, 8 vertices!
The proof, step by step
Prove that the cube satisfies the Euler formula F + V − E = 2.
- Euler's Formula states that for any convex polyhedron, F + V - E = 2.
- For our cube, F = 6, V = 8, E = 12.
- Substituting values: 6 + 8 - 12 = 2. The formula holds true.
Worked example
A polyhedron has 6 faces and 8 vertices. How many edges does it have?
Using Euler's formula F + V - E = 2, we get 6 + 8 - E = 2, so E = 12.
- 10
- 12 — correct
- 14
- 16