Cuboid Calculator
Enter the length, width and height of a cuboid (rectangular box) to instantly compute its volume, total surface area, lateral surface area, base area, space diagonal and all three face diagonals. Switch between metric and imperial units at any time. The "Show your work" panel walks through every formula so you can follow along or check a homework problem.
What is a cuboid?
A cuboid (also called a rectangular prism or rectangular parallelepiped) is a three-dimensional solid with six rectangular faces, twelve edges, and eight vertices. Every angle in a cuboid is a right angle, and opposite faces are identical rectangles. A cube is simply a special cuboid in which all three dimensions are equal. You encounter cuboids daily in the form of boxes, books, bricks, rooms, shipping cartons, and almost any everyday rectangular object.
How each property is calculated
Let a, b and c stand for the length, width and height of the cuboid. Volume measures the space enclosed: V = a x b x c. Doubling any one dimension doubles the volume. Total surface area adds the areas of all six faces: A = 2(ab + ac + bc). The factor of 2 accounts for the pair of identical opposite faces. Lateral surface area counts only the four side walls, excluding the top and bottom: Al = 2c(a + b). This is useful when calculating how much material covers the sides of a box. The space diagonal is the longest straight line you can draw inside the cuboid, from one corner to the opposite corner: D = sqrt(a squared + b squared + c squared). This follows directly from applying the Pythagorean theorem twice in succession. Face diagonals cross a single rectangular face and follow the same principle with two dimensions at a time.
Practical uses
Volume is the key figure for capacity problems: how many litres fit in a tank, how many cubic metres of concrete a slab needs, or how much product fills a shipping box. Surface area drives material-estimation problems: how much cardboard wraps a package, how much paint covers a wall, or how much insulation is needed. The space diagonal matters when checking whether an object (such as a long rod or a diagonal brace) will fit inside a rectangular enclosure. Face diagonals appear in carpentry and engineering when cutting a diagonal brace or checking the squareness of a frame.
Worked example
Suppose a cardboard box measures 30 cm long, 20 cm wide and 15 cm tall. Volume = 30 x 20 x 15 = 9,000 cm cubed (9 litres). Total surface area = 2(30 x 20 + 30 x 15 + 20 x 15) = 2(600 + 450 + 300) = 2 x 1,350 = 2,700 cm squared. Lateral surface area = 2 x 15 x (30 + 20) = 30 x 50 = 1,500 cm squared. Space diagonal = sqrt(30 squared + 20 squared + 15 squared) = sqrt(900 + 400 + 225) = sqrt(1,525) = about 39.06 cm. So you would need at least 2,700 cm squared of cardboard to fully cover the outside, but only 1,500 cm squared if you leave the top and bottom open.
Cuboid properties at a glance
| Property | Formula | Notes |
|---|---|---|
| Volume | V = a × b × c | Cubic units |
| Total Surface Area | A = 2(ab + ac + bc) | Sum of all 6 faces |
| Base Area | Ab = a × b | One top/bottom face |
| Lateral Surface Area | Al = 2c(a + b) | 4 side faces only |
| Space Diagonal | D = sqrt(a² + b² + c²) | Corner to opposite corner |
| Face Diagonal (a-b) | d1 = sqrt(a² + b²) | Across the base |
| Face Diagonal (a-c) | d2 = sqrt(a² + c²) | Across a side face |
| Face Diagonal (b-c) | d3 = sqrt(b² + c²) | Across the other side face |
| Edges | 12 total | 4 of each length a, b, c |
| Vertices | 8 | Each shared by 3 faces |
| Faces | 6 | 3 pairs of identical rectangles |
Key formulas for a cuboid with dimensions a (length), b (width), and c (height).
Frequently asked questions
What is the difference between a cuboid and a cube?
A cube is a special case of a cuboid in which all three dimensions (length, width and height) are equal. Every cube is a cuboid, but not every cuboid is a cube. In a cuboid the three dimensions can be any positive values, so the faces can be different rectangles. In a cube all six faces are identical squares.
How do I find the space diagonal of a cuboid?
The space diagonal D is found by applying the Pythagorean theorem in three dimensions: D = sqrt(a squared + b squared + c squared), where a, b and c are the length, width and height. This gives the length of the line from one corner of the box to the diagonally opposite corner through the interior.
What is the difference between surface area and lateral surface area?
Total surface area includes all six faces of the cuboid. Lateral surface area includes only the four side faces and excludes the top and bottom. If you are estimating how much paint is needed to cover the walls of a rectangular room (but not the floor or ceiling), the lateral surface area is the figure you want.
Can I use this calculator to find a missing dimension?
This calculator solves forward: you enter the three dimensions and it computes volume, surface area and diagonals. To find a missing dimension, rearrange the relevant formula. For example, if you know the volume V and two dimensions a and b, the height is c = V / (a x b). If you know the surface area and two dimensions, substitute into A = 2(ab + ac + bc) and solve for the third.
What units does this calculator use?
You can work in centimetres or inches by choosing Metric or Imperial from the units drop-down. All outputs (volume, area, diagonal) are expressed in the corresponding cubic, square or linear version of whichever unit you choose.
How do I convert volume from cubic centimetres to litres?
One litre equals exactly 1,000 cubic centimetres, so divide the volume in cm cubed by 1,000 to get litres. For example, a box with dimensions 30 cm x 20 cm x 15 cm has a volume of 9,000 cm cubed = 9 litres.