275. The Cube's Diagonal · always √3 times the edge

Try to break it

Drag A to resize the cube. The edge length a changes, and the space diagonal always comes out as a√3. Halve the cube (a → a/2) and the diagonal halves too. The √3 factor stays the same for every cube — that's the cube's signature ratio.

How you build it

Construct the cube, then draw the face diagonal AC and the space diagonal AG.

• Square tool: draw the front face (click a corner, then drag to size).
• Point tool: mark A at the top-left corner of the front square.
• Point tool: mark B at the top-right corner.
• Point tool: mark C at the bottom-right corner.
• Point tool: mark D at the bottom-left corner.
• Square tool: draw the back face the same size, offset up and to the right.
• Point tool: mark E at the back corner above A.
• Point tool: mark F at the back corner above B.
• Point tool: mark G at the back corner above C.
• Point tool: mark H at the back corner above D.
• Segment tool: join A to E.
• Segment tool: join B to F.
• Segment tool: join C to G — perpendicular to the front face.
• Segment tool: join D to H — the cube is complete.
• Segment tool: join A to C — the front-face diagonal. AC² = a² + a² = 2a².
• Segment tool: join A to G — the space diagonal. CG ⊥ the front face, so the right angle is at C: AG² = AC² + CG² = 2a² + a² = 3a², giving AG = a√3.

The proof, step by step

Prove that the space diagonal of a cube is √3 times its edge.

• Consider the bottom face OABC. The face diagonal OB has length a√2 by Pythagoras theorem.
• Now consider the right triangle OBG formed by the base diagonal OB, vertical edge BG, and space diagonal OG.
• Apply Pythagoras again: OG² = OB² + BG² = (a√2)² + a² = 2a² + a² = 3a².
• Therefore, OG = a√3. The space diagonal is always √3 times the edge length.

Worked example

A cube has an edge length of 6 cm. Find the length of its space diagonal.

Using the formula for the space diagonal of a cube, d = a√3. Substituting a = 6 cm, we get d = 6√3 cm.

• 6√2 cm
• 6√3 cm — correct
• 18 cm
• 36 cm