A rectangular prism, which is to say a box, has a volume equal to the product of its three dimensions and a surface area equal to twice the sum of the three face areas. Both are simple, but they scale differently: doubling every dimension multiplies the volume by eight and the surface area only by four.
That divergence is the entire reason bulk shipping is cheaper per unit than small parcels. A carton twice the size in every direction holds eight times as much while needing only four times the cardboard. The space diagonal is the longest object that will fit inside, which is the number that actually decides whether an awkward item will go in the box.
The common carton sizes in centimetres table follows the calculator: 7 rows spanning 10 x 10 x 10 through 100 x 50 x 50, there so the result can be cross-checked against a value you already trust. The page opens with length already set to 30 cm, so there is a finished result on screen before anything is typed. The result is also read against 4 bands, from small box at the low end to very large box at the high end, each with a note on what that range means in practice. The pages people most often open next are Torus Volume Calculator and Pyramid Volume Calculator.
The formula
l- Length
w- Width
h- Height
d- Space diagonal, √(l² + w² + h²)
How it works, step by step
- Enter the three dimensions in the same units.
- The gauge shows the volume.
- Surface area and space diagonal appear below.
- The litre figure assumes centimetre inputs.
Worked examples
A 30 by 20 by 15 cm carton
Volume is 9,000 cubic cm, or 9 litres. Surface area is 2 × (600 + 450 + 300) = 2,700 square cm, and the space diagonal is 39.051 cm.
Doubling every dimension
A 60 by 40 by 30 carton holds 72,000 cubic cm, eight times as much, while using 10,800 square cm of board — only four times more.
Checking the 40 x 30 x 20 row
The reference table lists Dimensions = 40 x 30 x 20, giving Volume (cu cm): 24,000, Litres: 24.00, Surface area: 5,200, Diagonal: 53.85. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
How do I convert to litres?
Divide cubic centimetres by 1000. A box measuring 30 by 20 by 15 cm holds nine litres.
What is the space diagonal for?
It is the longest straight item that will fit inside, corner to opposite corner. For 30 by 20 by 15 that is about 39.05 cm.
Why does bigger packaging use less material per unit?
Because volume scales with the cube of size and surface area only with the square. The ratio of area to volume falls as the box grows.
Does the order of the three dimensions matter?
Not for volume, surface area or diagonal — all three are symmetric in the inputs. It only matters for labelling which face is which.
What do the bands under the volume mean?
They label the figure so it can be read without a reference point of your own: Small box from 0 to 2,000; Medium box from 2,000 to 10,000; Large box from 10,000 to 50,000; Very large box from 50,000 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Rectangular Prism Volume Calculator use?
It computes V = l × w × h; A = 2(lw + lh + wh), where l is length; w is width; h is height; d is space diagonal, √(l² + w² + h²). The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Box volumes and surface areas
| Dimensions | Volume (cu cm) | Litres | Surface area | Diagonal |
|---|---|---|---|---|
| 10 x 10 x 10 | 1,000 | 1.00 | 600 | 17.32 |
| 20 x 15 x 10 | 3,000 | 3.00 | 1,300 | 26.93 |
| 30 x 20 x 15 | 9,000 | 9.00 | 2,700 | 39.05 |
| 40 x 30 x 20 | 24,000 | 24.00 | 5,200 | 53.85 |
| 50 x 40 x 30 | 60,000 | 60.00 | 9,400 | 70.71 |
| 60 x 40 x 40 | 96,000 | 96.00 | 12,800 | 82.46 |
| 100 x 50 x 50 | 250,000 | 250.00 | 25,000 | 122.47 |
Volume rises far faster than surface area as the carton grows.