Rectangular Prism Calculator
A 2 by 3 by 4 rectangular prism has a volume of 24 cubic units and a total surface area of 52 square units. This rectangular prism calculator finds volume, total surface area, lateral surface area, and the corner-to-corner space diagonal from length, width, and height. A rectangular prism, also called a cuboid, has six rectangular faces and right angles at every corner. Volume describes interior capacity, total surface area describes all six outer faces, and lateral area excludes the top and bottom.
Quick answer
Volume is the space inside the prism, found by multiplying length, width, and height.
What this tells you
- •Volume is the space inside the prism, found by multiplying length, width, and height.
- •Surface area is the total area of all six faces.
- •The space diagonal is the straight-line distance between two opposite corners.
- •Results are rounded to 4 decimal places.
- •Lateral surface area covers the four side faces and excludes the length-by-width top and bottom.
- •Length, width, and height may be decimals, but all three must be positive and use one consistent unit.
- •Volume uses cubic units, surface values use square units, and the diagonal uses the original linear unit.
How to Use
- 1Enter the prism length as a positive whole number or decimal.
- 2Enter the width and height in the same unit as the length. Convert mixed measurements before calculating.
- 3Calculate the results to see volume, total surface area, lateral surface area, and the space diagonal.
- 4Use volume for interior capacity and total surface area when every outside face must be covered.
- 5Use lateral surface area when only the four walls or side faces matter, with the top and bottom excluded.
- 6Attach the correct units. A prism measured in feet has cubic feet of volume, square feet of area, and a diagonal in feet.
How It Works
Formula
V = l x w x h, SA = 2(lw + lh + wh), diagonal = sqrt(l^2 + w^2 + h^2)Volume multiplies the three dimensions. Total surface area adds the three different face areas, length times width, length times height, and width times height, then doubles that sum. Lateral surface area is 2 x height x (length + width), which counts the four side faces but leaves out the top and bottom. The space diagonal applies the Pythagorean theorem in three dimensions. For a 2 by 3 by 4 prism, volume is 24, total surface area is 52, lateral area is 40, and the diagonal is sqrt(29) = 5.3852.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Volume and surface area of a 2 x 3 x 4 prism
Volume is 2 x 3 x 4 = 24. Surface area is 2 x (2 x 3 + 2 x 4 + 3 x 4) = 52. Lateral area is 2 x 4 x (2 + 3) = 40. The diagonal is sqrt(4 + 9 + 16) = 5.3852.
A cube with 5-unit sides
A cube is a rectangular prism with equal sides. Volume is 125 and total surface area is 6 x 25 = 150. Lateral area is 100, and the space diagonal is sqrt(75) = 8.6603.
Storage box measuring 10 x 4 x 2
Volume is 10 x 4 x 2 = 80. Total surface area is 2 x (40 + 20 + 8) = 136. Lateral area is 2 x 2 x (10 + 4) = 56. The diagonal is sqrt(120) = 10.9545.
Decimal dimensions of 1.5 x 2.5 x 6
Volume is 1.5 x 2.5 x 6 = 22.5. Total surface area is 2 x (3.75 + 9 + 15) = 55.5. Lateral area is 48. The diagonal is sqrt(44.5) = 6.6708.
Room-shaped prism measuring 12 x 8 x 3
Volume is 12 x 8 x 3 = 288. Total surface area is 2 x (96 + 36 + 24) = 312. Lateral area is 120. The diagonal is sqrt(217) = 14.7309.
Example Rectangular Prisms
Volume and surface area for sample dimensions.
| Length x Width x Height | Volume | Surface Area |
|---|---|---|
| 2 x 3 x 4 | 24 | 52 |
| 5 x 5 x 5 | 125 | 150 |
| 10 x 4 x 2 | 80 | 136 |
| 1 x 1 x 1 | 1 | 6 |
The table omits units because the formulas work with any consistent length unit. Volume is cubic and surface area is square.
Choosing the right prism measurement
Use volume when the question asks how much three-dimensional space fits inside a box-shaped region. Packing capacity, room air volume, soil in a raised bed, and water in a perfectly rectangular tank are volume questions. Real containers may need an allowance for wall thickness or unused headspace.
Use total surface area when all six faces count, such as covering a closed box. Use lateral surface area when only the four side walls count. The lateral formula excludes the length-by-width faces at the top and bottom.
Use the space diagonal for the straight-line distance between opposite corners through the interior. It differs from a face diagonal, which crosses only one rectangular face and uses two dimensions rather than all three.
Common mistakes
- Confusing volume with surface area. Volume measures the space inside in cubic units, while surface area measures the outside faces in square units.
- Forgetting to use the same units for all three dimensions. Mixing inches and feet gives a wrong result, so convert everything to one unit first.
- Using the face diagonal instead of the space diagonal. The space diagonal goes corner to corner through the prism and uses all three dimensions, not just two.
- Reporting volume in square units or surface area in cubic units. Volume is cubic, while both surface-area results are square.
- Using total surface area when the problem asks for lateral area. Total area includes the top and bottom, while lateral area does not.
- Subtracting material thickness without a clear reason. The formulas use the dimensions entered and do not infer inside measurements from outside measurements.
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