116. The Cuboid Code · 6 faces, 12 edges, 8 vertices — always

Try to break it

Drag point P to stretch or shrink the cuboid's depth. Try to make it lose a face, edge, or vertex along the way. You can't — a cuboid always has 6 faces, 12 edges, and 8 vertices, no matter how skinny or chunky it gets.

How you build it

Build a cuboid skeleton from two equal rectangles.

• Draw the FRONT face rectangle — click the front-top-left corner, then the front-bottom-right.
• Draw the BACK face rectangle the SAME size — click its top-left corner (above and right of the front face), then its bottom-right.
• Draw the depth edge from the front-top-left corner to the back-top-left corner.
• Draw the depth edge from the front-top-right corner to the back-top-right corner.
• Draw the depth edge from the front-bottom-right corner to the back-bottom-right corner.
• Draw the last depth edge from the front-bottom-left to the back-bottom-left corner — cuboid skeleton complete.

The proof, step by step

Prove that the cuboid has 6 faces, 12 edges, and 8 vertices.

• A cuboid has exactly six rectangular faces. Each pair of opposite faces is equal and parallel.
• A cuboid has exactly twelve edges. These are grouped into three sets of four parallel edges of equal length.
• A cuboid has exactly eight vertices (corners). At each vertex, three edges meet at right angles.
• Euler's formula for polyhedra holds: V - E + F = 2. For a cuboid: 8 - 12 + 6 = 2. This balance never breaks.

Worked example

A cuboid has dimensions 10 cm × 8 cm × 5 cm. How many edges does this cuboid have?

Every cuboid, regardless of its specific dimensions, always has exactly 12 edges. The dimensions only change the lengths of these edges, not their count.

• 8
• 10
• 12 — correct
• 16