How this instrument works
A cube has exactly one free measurement — the side length s — because all twelve edges are forced to match it. Everything else you might want to know about the cube is locked in as soon as s is: the space it holds (V = s³), the material needed to skin it (SA = 6s²), and the longest straight line that fits inside it, corner to opposite corner (d = s√3). This sheet exists because those three questions come up together constantly — a packing problem wants capacity and a diagonal check, a fabrication job wants surface area and volume — and retyping the same side length into three separate calculators is pure friction.
The diagonal is the one figure most people don't already carry in their head, and it comes from the Pythagorean theorem applied twice. First, the diagonal across one square face spans s√2. Then treat that face diagonal and the vertical edge as the two legs of a second right triangle standing up through the cube's interior: the space diagonal is the hypotenuse, √(s² + (s√2)²) = √(3s²) = s√3. Two flat calculations, stacked, give a genuinely three-dimensional length.
Watch how differently the three outputs grow. Volume answers to the cube of s, surface area to its square, and the diagonal — like the side itself — only to the first power: double s and the diagonal exactly doubles too, even while volume grows eightfold. At the far edge, s = 0 shrinks the whole solid to a single point, and all three outputs fall to zero together without the formulas needing a special case.
- Type your cube's edge measurement into the Side length field — any unit works, and results follow it (squared for area, unchanged for the diagonal).
- Read Volume for the space the cube encloses.
- Read Surface area for the total area of all six faces combined.
- Read Space diagonal for the straight-line distance from one corner through the interior to the opposite corner.
- Change Side length and all three outputs update together, since every one of them is computed straight from that single figure.
Worked example — three answers from side length 3
A cubic shipping crate is built with a side length of 3 feet. Volume: V = 3³ = 27 cubic feet, the capacity to fill or load. Surface area: SA = 6 × 3² = 6 × 9 = 54 square feet, the material figure to hand a supplier for the six panels. Space diagonal: d = 3√3 ≈ 5.196152 feet, the true corner-to-corner distance through the crate's interior — the number that decides whether a 5-foot rod packed diagonally will fit, since it clears the crate's straight edges by a comfortable margin but a rod any longer than about 5.196 feet will not.
Notice the diagonal is longer than the edge (3) and longer than a single face's diagonal (3√2 ≈ 4.243) but still shorter than the sum of two edges — it has to be, since it's the hypotenuse of a right triangle built from that face diagonal and one more edge standing perpendicular to it. All three results — 27, 54, and 5.196152422706632 — come from the one number typed into Side length, with nothing else to enter.
Questions
What are the formulas for a cube's volume, surface area, and diagonal?
Volume is V = s³, surface area is SA = 6s², and the space diagonal is d = s√3, where s is the length of one edge. All three are functions of that single number, which is why one input can drive all three results at once.
Why combine volume, surface area, and diagonal on one page instead of three?
Because a cube has only one independent measurement to begin with. Once s is fixed, volume, surface area, and diagonal are all already determined — computing them separately means typing the same side length into three different tools for no extra information.
How is the space diagonal formula d = s√3 derived?
From two applications of the Pythagorean theorem. A face diagonal spans s√2 across one square face; treating that face diagonal and the perpendicular edge as legs of a second right triangle gives a hypotenuse of √(s² + (s√2)²) = √(3s²) = s√3.
Is the space diagonal the same as a face diagonal?
No, and mixing them up is the most common error. A face diagonal (s√2 ≈ 1.414s) crosses one flat square face; the space diagonal (s√3 ≈ 1.732s) runs through the cube's interior from one corner to the opposite corner and is the longer of the two.
How do volume, surface area, and diagonal scale differently as the side grows?
Diagonal scales linearly with s, just like the side itself — doubling s doubles it. Surface area scales with s², so doubling s quadruples it. Volume scales with s³, so doubling s multiplies it by eight. Same input, three different growth rates.
What do the results mean when side length is zero?
All three outputs fall to zero together: a point has no volume, no surface, and no diagonal to measure. It's the fully degenerate limit of a cube rather than a shape you would meet in practice, but the formulas handle it cleanly with no special-casing required.