A rectangular prism is the shape of almost every box, room, tank, crate and slab you will ever measure. It has six rectangular faces, twelve edges and three independent dimensions, and this rectangular prism calculator turns those three numbers into volume, total surface area, lateral area, the internal space diagonal and the capacity in litres and US gallons. It also runs the useful reverse case: give it a target volume and two sides, and it tells you the height you need.
Arb Digital built this page because the arithmetic is easy and the mistakes are not. Almost every wrong answer in this area comes from mixing units, forgetting that a lidless box has five faces rather than six, or reaching for the wrong diagonal. The tool shows its working with your own numbers substituted in, so the result is checkable rather than just printed.
What This Rectangular Prism Calculator Does
In its default mode the calculator takes a length, a width and a height in a single unit and returns five things. The headline figure is volume, the product of the three dimensions. Below it you get total surface area, the sum of all six faces. Lateral area is the four vertical sides only, excluding the top and bottom, which is the number you want when you are painting the walls of a room or wrapping a label around a carton. The space diagonal is the longest straight line that fits inside, and the height tile confirms which height the calculation used, which matters in the reverse mode.
Switch the mode selector and the height field is replaced by a volume field. The calculator then divides your volume by length times width to recover the height. That is the everyday version of the problem: you know a tank must hold a certain number of litres and you know its footprint, so how deep does it need to be?
Capacity is reported alongside the geometry. Because a cubic centimetre is exactly one millilitre, the conversion to litres is a division by 1,000, and the conversion from litres to US gallons is a division by 3.785411784. Those two figures are usually what people actually wanted when they typed a box size into a search bar.
How to Use It
- Pick your mode. Leave it on the default if you have all three dimensions. Switch it if you know the volume and need the third side.
- Enter length and width. These two fields are used in both modes and must share the same unit.
- Enter the height or the volume. The form swaps the field automatically so only the relevant one is on screen.
- Choose your unit. Everything is reported back in that unit, with squared units for area and cubed units for volume.
- Read the working line under the results. It restates the formula with your figures in it so a mistyped digit is obvious.
The Formula: How a Rectangular Prism Is Calculated
With length l, width w and height h, the four core results are:
- Volume: V = l × w × h. For a 60 × 40 × 30 cm box, V = 72,000 cm³, which is 72 litres or about 19.02 US gallons.
- Total surface area: A = 2(lw + lh + wh). Three distinct face shapes, each appearing twice. For the same box: 2(2,400 + 1,800 + 1,200) = 10,800 cm².
- Lateral area: Aₗ = 2h(l + w). The four sides without the top and bottom. Here: 2 × 30 × 100 = 6,000 cm².
- Space diagonal: D = √(l² + w² + h²). For the same box: √(3,600 + 1,600 + 900) = √6,100 = 78.10 cm.
The space diagonal is the three-dimensional extension of the Pythagorean theorem, and it is worth seeing why. First find the diagonal across the base, √(l² + w²). Then treat that base diagonal and the vertical height as the two legs of a second right triangle. Squaring and adding gives l² + w² + h² under one root. The standard treatment is set out in Wolfram MathWorld's entry on the rectangular parallelepiped.
Why the Cube Is the Cheapest Box
Here is a result that surprises people: among all rectangular prisms with the same volume, the cube has the smallest total surface area. Since packaging cost tracks surface area and shipping cost tracks volume, the closer a box is to cubic, the less material it uses for the space it encloses.
Run the numbers. A box of 72 litres shaped 60 × 40 × 30 cm has a surface area of 10,800 cm². The same 72,000 cm³ arranged as a cube has an edge of about 41.6 cm and a surface area of about 10,383 cm². That is roughly 4% less material for identical capacity. Stretch the box to 120 × 30 × 20 cm and the area climbs to 13,200 cm², more than 22% above the cubic minimum, for exactly the same contents.
The practical caveat is that packing efficiency, shelf depth and product shape usually dominate. A long thin product does not fit a cube. But when a box specification is genuinely free, moving the proportions toward equal sides is free money on materials. Taking the cube root of your target volume gives the cubic benchmark, so you can measure how far from optimal a proposed box really is with the cube root calculator.
The Open-Top Box Problem
The formula 2(lw + lh + wh) assumes a closed solid with six faces. A great many real objects are not closed. A planter, an open crate, a sink, an unlidded storage bin or an open-topped water tank has five faces, and the material you need is the total minus one lw face: lw + 2lh + 2wh.
For the 60 × 40 × 30 cm box, removing the top drops the surface area from 10,800 cm² to 8,400 cm², a 22% reduction. If you were pricing sheet metal or plywood from the closed-box figure, you would have overspent by nearly a quarter. Similarly, when you are tiling or lining the inside of a rectangular pit or pool, the top is open and the floor is a single lw rectangle, so the same five-face formula applies.
Rooms are the mirror image of this problem. To paint the walls of a room you want lateral area only, and then you subtract door and window openings from it. For the room-by-room version of that calculation, use the wall area calculator, which is built for exactly that subtraction.
Fitting Long Objects Into a Box
The space diagonal answers the question every shipper eventually asks: will this thing fit? A 60 × 40 × 30 cm carton has a space diagonal of 78.1 cm, so a thin rigid rod up to 78.1 cm long fits diagonally even though no single edge is longer than 60 cm. That is 30% more reach than the longest side.
Two cautions apply. First, the diagonal is a mathematical line with no thickness. Anything with real width or depth needs clearance at both corners, so the practical maximum is always shorter than the calculated diagonal, and the gap grows quickly as the object gets thicker. Second, an object that fits the diagonal cannot usually be rotated once it is in, which matters if it has to sit inside protective packaging.
The two-dimensional version of the same question, whether a picture or panel fits through a doorway, is the plain Pythagorean case handled by our Pythagorean theorem calculator.
Units, Capacity and What This Tool Is Not
Cubed units scale with the cube of the linear factor. One cubic metre is 1,000,000 cubic centimetres, not 100. One cubic foot is 1,728 cubic inches. One cubic yard is 27 cubic feet, which is the number that catches out anyone ordering bulk material in the United States. The SI definitions underlying these conversions are published by NIST.
It is worth being clear about the boundary between this page and two adjacent tools. This calculator computes a volume from physical dimensions. The volume converter does something different: it takes a volume you already have and restates it in another unit, with no geometry involved. Likewise, the concrete calculator starts from a slab or footing volume and turns it into bags, cubic yards and cost, so it picks up where this page finishes.
Arb Digital designs, writes and ships free calculator pages that rank, complete with schema, internal linking and the technical SEO underneath them.
Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Mixing units between dimensions. A length in feet with a height in inches produces a meaningless product. Convert everything first.
- Using the six-face formula on an open container. Drop one lw face for a lidless box, or you will over-order material by a fifth or more.
- Confusing lateral area with total area. Painting walls uses 2h(l + w). Wrapping a parcel uses the full 2(lw + lh + wh).
- Assuming the longest edge is the fitting limit. The space diagonal is longer, and it is the real internal constraint.
- Reading internal capacity from external dimensions. Wall thickness reduces usable volume, and on small thick-walled containers the difference is substantial.
Related Free Tools From Arb Digital
For round containers, the cylinder volume calculator covers tanks, pipes and drums. If your prism is a concrete pour, the concrete calculator converts the volume into materials, and the cost per square foot calculator handles the pricing side. Browse the full free online tools hub for the rest.
Frequently Asked Questions
Multiply length by width by height: V = l × w × h. A 60 × 40 × 30 cm box holds 72,000 cubic centimetres, which is 72 litres or about 19.02 US gallons.
Use A = 2(lw + lh + wh). The prism has three distinct face shapes and each appears twice. For a 60 × 40 × 30 cm box that gives 10,800 square centimetres.
Lateral area is the four side faces only, excluding the top and bottom, calculated as 2h(l + w). It is the figure you want when painting the walls of a room or wrapping a band around a carton.
Divide the volume by length times width. A 72,000 cubic centimetre volume on a 60 by 40 cm footprint needs a height of 30 cm. The second mode of this calculator does that step for you.
The space diagonal, calculated as the square root of l squared plus w squared plus h squared. A 60 by 40 by 30 cm carton has a diagonal of 78.1 cm, well beyond its longest edge of 60 cm.
Yes. Cuboid, rectangular parallelepiped and rectangular prism all describe a solid with six rectangular faces. A cube is the special case where all three dimensions are equal.
Among all rectangular prisms of equal volume, surface area is minimised when the three dimensions are equal. Stretching a box in one direction adds surface area without adding capacity.
Results are mathematical values rounded for display. Real containers have wall thickness, tolerances and waste allowances, so add a margin before cutting, ordering or pouring.
