Rectangular Box (Cuboid) Calculator
This calculator works out the volume, surface area, diagonals and edge length of any rectangular box, or cuboid, from three measurements. Enter the length, width and height, then choose your unit: metres, centimetres, millimetres, inches or feet, and every result updates instantly. You get the volume (how much space it encloses), the total surface area across all six faces, the space diagonal (the straight line running from one corner through the interior to the opposite corner), the three face diagonals for each pair of dimensions, and the total edge length across all twelve edges. A worked example using the default 5 x 4 x 3 cm box is shown alongside your own figures, so you can see exactly how each formula is applied: volume as length times width times height, surface area as 2(lw + lh + wh), and the space diagonal using a three-dimensional version of the Pythagorean theorem. Use this tool to work out how much soil, concrete or water a box-shaped space will hold, how much paint, wrapping paper or sheet material you need to cover a box, whether a rod or pipe will fit inside a container at an angle, or how much framing or edging trim is required to outline a rectangular shape. Just make sure all three dimensions are entered in the same unit before calculating, since mixing units will throw the results off.
Dimensions
Please enter positive numbers for all three dimensions.
Quick Results
Worked Example (defaults)
l = 5 cm, w = 4 cm, h = 3 cm
Volume = 5 x 4 x 3 = 60 cm³
SA = 2(5x4 + 5x3 + 4x3) = 2(20+15+12) = 94 cm²
Space diagonal = sqrt(25+16+9) = sqrt(50) = 7.071 cm
All Measurements
Diagonals and Edges
Cuboid Formulas
A rectangular box (cuboid) has six rectangular faces, twelve edges, and eight vertices. With dimensions length (l), width (w), and height (h), the standard formulas are:
| Property | Formula | Example (5 x 4 x 3) |
|---|---|---|
| Volume | V = l x w x h | 5 x 4 x 3 = 60 cm³ |
| Surface area | SA = 2(lw + lh + wh) | 2(20 + 15 + 12) = 94 cm² |
| Space diagonal | d = sqrt(l² + w² + h²) | sqrt(50) = 7.071 cm |
| Face diagonal (base l x w) | sqrt(l² + w²) | sqrt(41) = 6.403 cm |
| Face diagonal (front l x h) | sqrt(l² + h²) | sqrt(34) = 5.831 cm |
| Face diagonal (side w x h) | sqrt(w² + h²) | sqrt(25) = 5.000 cm |
| Total edge length | 4(l + w + h) | 4 x 12 = 48 cm |
Volume of a Rectangular Box
Volume measures the three-dimensional space enclosed by the box. You multiply length by width by height. The result is in cubic units: if your dimensions are in centimetres, your volume is in cubic centimetres (cm³). If your dimensions are in metres, your volume is in cubic metres (m³).
Common uses for volume calculations include working out how much soil, concrete, water, or air a box-shaped space holds. For example, a garden bed 2 m long, 1 m wide, and 0.3 m deep needs 0.6 m³ of soil to fill it.
Surface Area of a Cuboid
A cuboid has three pairs of identical rectangular faces. The total surface area adds all six faces: two l x w faces (top and bottom), two l x h faces (front and back), and two w x h faces (left and right). Surface area is in square units and is useful for calculating paint coverage, wrapping paper needed, or the amount of material required to build a box.
Space Diagonal
The space diagonal is the longest straight line you can draw inside a rectangular box, connecting two opposite corners through the interior. It is found using a three-dimensional extension of the Pythagorean theorem: d = sqrt(l² + w² + h²). This is useful when checking whether an object (such as a rod or pipe) will fit inside a box at an angle.
Each face also has its own shorter diagonal, calculated from the two side lengths of that face using the standard Pythagorean theorem.
Total Edge Length
A cuboid has twelve edges: four of each length l, w, and h. The total edge length is 4(l + w + h). This is useful when calculating the amount of framing timber, metal edging strip, or decorative trim needed to outline a rectangular box shape.
Related Calculators
- Maths and Stats Calculators: full list of geometry and maths tools.
- Cube Surface Area and Volume Calculator: for a box where l = w = h.
- Area Calculator: area of rectangles, circles, triangles, and more.
- Area of a Triangle Calculator: base and height or Heron's formula.
- Arc Length and Sector Area Calculator: circular geometry.
Method: Standard Euclidean geometry. Volume formula V = lwh; surface area SA = 2(lw + lh + wh); space diagonal d = sqrt(l² + w² + h²); face diagonals via Pythagorean theorem on each pair of dimensions; total edge length = 4(l + w + h). No rounding is applied internally; displayed values are rounded to three decimal places.
This calculator uses exact formulas for all outputs. Results are rounded to three decimal places for display. Ensure all dimensions are entered in the same unit before calculating.