Rectangular Prism Calculator
Calculate a box-shaped prism's volume, total surface area, and corner-to-corner space diagonal from three positive dimensions.
Dimensions
Use one unit for all three edges. Changing the unit converts the current dimensions.
Required. Positive decimal; use a period for decimals.
Required. Enter the second perpendicular edge.
Required. Enter the vertical or third perpendicular edge.
Area uses squared units; volume uses cubed units.
Live results
Results update as soon as all dimensions form a valid prism.
Volume
3,240 in³
Surface area
1,332 in²
Space diagonal (d)
26.325 in
A prism measuring 18 × 12 × 15 in has a volume of 3,240 in³.
Surface-area breakdown
Each rectangle appears twice on opposite sides of the prism.
| Face pair | Dimensions | One face area | Opposite-pair contribution |
|---|---|---|---|
| Length × width | 18 × 12 in | 216 in² | 432 in² |
| Length × height | 18 × 15 in | 270 in² | 540 in² |
| Width × height | 12 × 15 in | 180 in² | 360 in² |
| Total surface area | 1,332 in² | ||
The table is a direct decomposition of A = 2(lw + lh + wh). It is useful for estimating wrapping, paint, sheet material, or exposed exterior area when all six faces count.
How to use this rectangular prism calculator
What this calculator does
This calculator evaluates a right rectangular prism, also called a cuboid or box. From three mutually perpendicular edge lengths, it returns the enclosed Volume, the area of all six outside faces as Surface area, and the straight corner-to-opposite-corner Space diagonal (d). It also provides a face-pair table that decomposes the surface-area calculation. These are exact geometric identities for ideal rectangular dimensions; they do not account for wall thickness, seams, lids, openings, material waste, or measurement uncertainty.
When to use it
Use the calculator to estimate a storage box's capacity, compare aquarium or tank dimensions, calculate wrapping or coating area, check whether a rigid object fits diagonally inside a rectangular space, or verify homework and design calculations. For irregular containers or shapes with sloped sides, a rectangular-prism model may only be an approximation.
How to calculate
- The calculator opens with a complete demonstration: 18 in long, 12 in wide, and 15 in high. Its validated example workbook is immediately available through Download Excel.
- Replace Length (l), Width (w), and Height (h) with your measured values. Use positive numbers and a period as the decimal separator; grouping commas are accepted only in standard three-digit groups.
- Choose the shared Length unit. Changing it converts all three current dimensions so the physical prism remains the same. Results then use that unit, its square for area, and its cube for volume.
- Read the primary result and supporting cards, review the face-pair table, and select Download Excel to create a current-state OOXML workbook.
- Reset clears the demonstration and all calculated content. Download Excel is then disabled until a complete valid set of dimensions is entered again.
Input guide
Length (l) is a required positive decimal representing one edge, such as 18 in. A larger length increases volume linearly, increases two pairs of face areas, and lengthens the diagonal. Do not enter a unit symbol in the field. Width (w) is the required second perpendicular edge, such as 12 in, and affects the same output types in the corresponding width terms. Height (h) is the required third perpendicular edge, such as 15 in; for a box it is often the vertical measurement, but orientation does not change the formulas. Zero, negative values, scientific notation, mixed text, ambiguous decimal commas, and values above the supported finite range are rejected instead of being silently changed.
Length unit is a required selection shared by all dimensions. Available choices are millimeters, centimeters, meters, kilometers, inches, feet, and yards. For example, switching the startup dimensions from inches to feet converts them to 1.5, 1, and 1.25 ft. The geometry stays constant while the numeric area and volume values change to ft² and ft³. The NIST guide to length, area, and volume units explains why area uses squared units and volume uses cubed units.
Output guide
Volume measures enclosed three-dimensional space in cubic units and is calculated as l × w × h. Surface area measures all six faces in square units and equals 2(lw + lh + wh). Space diagonal (d) is a length, calculated with the three-dimensional Pythagorean relationship √(l² + w² + h²). The three header pills repeat Volume, Surface, and Diagonal for rapid scanning.
In the breakdown table, Face pair identifies the edge combination; Dimensions shows the two sides of that rectangle; One face area is their product; and Opposite-pair contribution doubles it because each face has a matching opposite face. A very small or large result is not inherently wrong, but it should be checked against the selected unit and the scale of the real object. Valid dimensions always produce positive results.
Worked example
With Length = 18 in, Width = 12 in, and Height = 15 in, volume is 18 × 12 × 15 = 3,240 in³. The three one-face areas are 216, 270, and 180 in²; doubling their sum gives 2 × (216 + 270 + 180) = 1,332 in². The diagonal is √(18² + 12² + 15²) = √693 = 26.325 in when displayed to three decimal places. These values match the first-open results and the startup workbook.
Learn more
For a textbook treatment of prism volume and surface area, review the OpenStax volume and surface-area formulas. A compact reference for the cuboid formulas, including face and space diagonals, is also available in Wolfram MathWorld's cuboid entry.
Formula notes and practical interpretation
Volume grows in direct proportion to each individual dimension. Holding width and height fixed while doubling length doubles volume. If all three dimensions are doubled, volume becomes eight times as large, surface area becomes four times as large, and the diagonal becomes twice as long. This scaling difference is important when comparing packaging, rooms, tanks, or storage bins: capacity can grow much faster than exterior material requirements.
Surface area assumes every face is included. For an open-top container, subtract the missing face. For paint or cladding, also subtract doors, windows, vents, and other uncovered regions, then add a waste allowance separately. Volume likewise describes ideal interior space only when the entered dimensions are interior measurements; exterior dimensions of a thick-walled container overstate usable capacity.
The space diagonal is the longest straight segment between two vertices of a right rectangular prism. It can help with fit checks, but real objects may need clearance for rotation, packaging, handles, or irregular profiles. Treat the result as a geometric limit rather than a guarantee that an object can be maneuvered through an opening.