Volume, surface area, and space diagonal of any box (or cube)
Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 12, 2026
A rectangular prism (also called a cuboid or box) has three pairs of opposite rectangular faces. Length l, width w, and height h are the three different edge lengths. A cube is the special case where l = w = h. These three dimensions give you everything: volume, surface area, and the longest internal diagonal.
| Name | Formula | Notes |
|---|---|---|
| Volume | V = l × w × h |
l = length, w = width, h = height. Cubic units (cm³, m³, …). |
| Total Surface Area | SA = 2 × (lw + lh + wh) |
Sum of all 6 face areas. Three distinct face shapes, each appearing twice. |
| Lateral Surface Area | SA_lat = 2h × (l + w) = Ph |
Just the four side faces (no top/bottom). P is the base perimeter. |
| Space Diagonal | d = √(l² + w² + h²) |
Longest internal line — corner to opposite corner through the interior. The 3D Pythagorean theorem. |
| Cube Special Case | V = s³, SA = 6s², d = s√3 |
When l = w = h = s, all three formulas simplify to single-variable forms. |
| Height from Volume | h = V / (l × w) |
Inverse — find one dimension when you know the volume and the other two. |
| Edge Sum (all 12 edges) | E = 4(l + w + h) |
A box has 12 edges in 3 groups of 4 parallel edges. |
Plug in your numbers and get instant step-by-step results.