Box Diagonal Calculator
From a length, a width, and a height, get the space diagonal, the volume, and the surface area — the numbers that describe any rectangular box.
Three inputs, three answers
Enter the length, width, and height and the calculator returns the space diagonal (√(l²+w²+h²)), the volume (l·w·h), and the surface area (2·(lw+lh+wh)) at once.
Keep units consistent
The three edges are unit-agnostic — your answers come back in the same unit (squared for the surface, cubed for the volume), so don't mix centimetres with inches.
What is a box diagonal calculator?
Three edges in, full box out
A box diagonal calculator turns three measurements — the length, width, and height of a rectangular box (a cuboid) — into the numbers that describe the whole solid: the space diagonal (the longest straight line you can draw inside it, from one corner to the opposite corner), the volume (how much it holds), and the surface area (the outside of all six faces). The space diagonal is the headline figure: it tells you the longest object that could fit inside a box, or the shortest straight run between two opposite corners. That makes it the number to know when checking whether a pole, a shelf, or a screen will slide into a carton, a drawer, or a shipping crate.
Enter the length, width, and height in any length unit to get the space diagonal, volume, and surface area instantly.
Three short formulas, all built from the three edge lengths alone — no constants needed.
diagonal = √(l² + w² + h²)The space diagonal extends the Pythagorean theorem into three dimensions: square each edge, add them, take the square root. The volume is simply length × width × height — the box packed solid. The surface area adds the three distinct pairs of faces: 2 × (l·w + l·h + w·h), counting top and bottom, front and back, left and right.
Suppose you have a box that is 3 long, 4 wide, and 12 tall.
Space diagonal
√(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 — the longest line inside (a clean 3-4-12-13 Pythagorean quadruple).
Volume
3 × 4 × 12 = 144 cubic units — the space inside.
Surface area
2 × (3·4 + 3·12 + 4·12) = 2 × (12 + 36 + 48) = 192 square units — all six faces.
The three outputs answer three different everyday questions. The space diagonal (13 for a 3 × 4 × 12 box) is the most useful insight: it is the longest straight object that can fit inside the box, so a 12.9-unit pole slides in but a 13.1-unit one will not, no matter how you angle it. It is always longer than any single edge, because it spans all three dimensions at once. The volume (144 cubic units here) is how much the box holds — the packing capacity, the displaced water, the material if it were solid. The surface area (192 square units) is the outside of all six faces, the figure you need to wrap, paint, or estimate cardboard for. Notice the diagonal grows much more slowly than the volume: doubling every edge multiplies the volume by eight but the diagonal by only two, which is why a slightly bigger box holds far more without becoming much harder to fit through a doorway.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Right-angled boxes and consistent units
These formulas describe a perfect rectangular box (cuboid) with all edges meeting at right angles. A slanted or skewed box, a box with rounded edges, or one whose walls have real thickness will differ from the computed value. The three edges are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: edges in centimetres give a diagonal in centimetres, a surface in square centimetres, and a volume in cubic centimetres, never a mix.