A rectangular prism — also called a cuboid or simply a box — is the most common 3-D shape in everyday life: rooms, shipping cartons, aquarium tanks, concrete footings, timber beams. This calculator gives you its volume, total surface area, space diagonal and all three face perimeters from just three measurements, and can equally work backwards to find a missing dimension when you know the volume.
How it works
Enter the three dimensions — length (l), width (w) and height (h) — and choose a unit. Results update as you type, so you can explore different configurations instantly.
Solve for a missing dimension by switching the “Solve for” dropdown away from Volume. Select which side is unknown, type in the known volume and the two remaining sides, and the calculator rearranges the formula to isolate the missing measurement.
An isometric diagram updates dynamically to reflect the proportions of your prism, with each axis colour-coded so you can immediately see which dimension maps to which label.
The formulas
All calculations use exact closed-form arithmetic — no approximations.
Volume:
V = l x w x h
Total surface area (sum of the six rectangular faces):
SA = 2 x (l x w + l x h + w x h)
Space diagonal (longest internal straight line, corner to corner):
d = sqrt(l squared + w squared + h squared)
The space diagonal formula comes from applying the Pythagorean theorem twice: first across the base to get the face diagonal sqrt(l squared + w squared), then up the height to extend it into three dimensions.
Reverse-solve for any dimension:
l = V / (w x h) | w = V / (l x h) | h = V / (l x w)
Worked example
A storage crate measures 1.2 m long, 0.8 m wide and 0.6 m tall.
- Volume = 1.2 x 0.8 x 0.6 = 0.576 m cubed (576 litres)
- Surface area = 2 x (1.2 x 0.8 + 1.2 x 0.6 + 0.8 x 0.6) = 2 x (0.96 + 0.72 + 0.48) = 2 x 2.16 = 4.32 m squared
- Space diagonal = sqrt(1.44 + 0.64 + 0.36) = sqrt(2.44) ≈ 1.562 m
Now imagine you need to fit exactly 1 m cubed of material into a crate that is 1.25 m long and 0.8 m wide. Switch to “Solve for Height”: h = 1 / (1.25 x 0.8) = 1.0 m tall. The crate needs to be exactly 1 metre high.
| l (m) | w (m) | h (m) | Volume (m³) | Surface area (m²) |
|---|---|---|---|---|
| 1.0 | 1.0 | 1.0 | 1.000 | 6.000 |
| 2.0 | 1.0 | 0.5 | 1.000 | 7.000 |
| 1.2 | 0.8 | 0.6 | 0.576 | 4.320 |
| 0.3 | 0.2 | 0.1 | 0.006 | 0.220 |
Notice the first two rows have equal volume but different surface areas — a cube minimises surface area for a given volume, which is why manufacturers favour nearly-cubic packaging.
Formula note
The volume formula V = l x w x h is a special case of Cavalieri’s principle: the volume of any prism equals its base area multiplied by its perpendicular height. For a rectangular base the base area is simply l x w, giving V = (l x w) x h. The surface-area formula follows directly from counting the three distinct pairs of congruent faces: two of area l x w, two of l x h, and two of w x h.