Alliance Form 3 P1 Q13 — Diagonal of a Cube Face
Published
The Question
“A cube has a volume of 1,728 cubic centimetres. Find the length of the diagonal across one of its faces, giving your answer to two decimal places.”
Find the edge from the volume
Every edge of a cube is the same length, and the volume equals that edge length multiplied by itself three times. So to undo the cubing and recover the edge, take the cube root of the volume.
Look at a single square face
One face of the cube is a flat square measuring twelve centimetres by twelve centimetres. The diagonal we want runs from one corner of this square to the opposite corner, cutting the square into two right-angled triangles.
Apply Pythagoras to the triangle
In each right-angled triangle the two edges of the square are the shorter sides and the diagonal is the hypotenuse. By Pythagoras, the square of the diagonal equals the sum of the squares of the two edges.
Evaluate and round
Work out the square root of two hundred and eighty-eight, then round to two decimal places as the question asks.
Final Result
The diagonal across one face of the cube is approximately 16.97 centimetres.
Why this method works
The method chains two ideas together. First, volume being the edge cubed means the cube root is the exact inverse operation that pulls the edge back out, giving a clean 12 cm. Second, a face diagonal is nothing more than the hypotenuse of a right-angled triangle formed by two equal edges, so Pythagoras applies directly. Because both short sides are equal, the diagonal is always the edge multiplied by the square root of two, which is why the answer comes out as the square root of 288.
Check: 12 times the square root of 2 is 12 × 1.41421 ≈ 16.97 cm, and 16.97 squared ≈ 288, matching the two 144s.