Space diagonal of a cube

From an edge a, the face diagonal is a√2 and the space diagonal is a√3.

Key formulas

On the base, the face diagonal is √(a²+a²)=a√2. Then the space diagonal joins that face diagonal with the vertical edge: √((a√2)²+a²)=a√3.

Edge 5 cm. Space diagonal = 5√3 cm. Do not stop at 5√2 — that is only across one face.

“Longest stick that fits inside the cube” usually means the space diagonal.

Problem. A cube has edge 4 cm. Find the length of a space diagonal.

Solution. Space diagonal = a√3 = 4√3 cm.

a = 2 → space diagonal 2√3.

Cube edge 5 cm. Space diagonal is:

Yes: corners (0,0,0) and (a,a,a) give distance a√3.