Rectangular Prism Calculator
Enter the length, width, and height of a rectangular prism (box) to instantly calculate its volume, total surface area, lateral surface area, face areas, and space diagonal. Choose metric or imperial units, and see the full step-by-step working below.
Formula
Worked example
A box measuring 10 cm x 6 cm x 4 cm: Volume = 10 x 6 x 4 = 240 cm^3. Surface area = 2(60 + 40 + 24) = 2 x 124 = 248 cm^2. Space diagonal = sqrt(100 + 36 + 16) = sqrt(152) = 12.329 cm.
What is a rectangular prism?
A rectangular prism, also called a cuboid, is a three-dimensional solid with six rectangular faces. Every pair of opposite faces is identical and parallel, all interior angles are 90 degrees, and there are 12 edges in three groups of four parallel edges. Everyday examples include shipping boxes, bricks, books, and rooms. When all three dimensions are equal the shape becomes a special case called a cube. The term "box" is informally used for the same shape in engineering and packaging.
Volume, surface area and space diagonal explained
The volume V = l x w x h tells you how much space is enclosed, measured in cubic units. The total surface area S = 2(lw + lh + wh) is the combined area of all six faces, important for painting, wrapping, or heat-loss calculations. The lateral surface area omits the top and bottom faces, which is useful when the two ends are open (like a tube or a room with no ceiling and floor in the calculation). The space diagonal d = sqrt(l^2 + w^2 + h^2) is the longest straight line that fits entirely inside the box, from one corner to the diagonally opposite corner. It tells you the maximum length of an object that can fit inside the box without bending.
Metric and imperial units
This calculator accepts dimensions in any consistent unit. In metric mode the inputs are in centimetres, giving volume in cm^3 and areas in cm^2. Switch to imperial to enter inches, receiving volume in in^3 and areas in in^2. If you need different units (metres, feet, millimetres) just enter the values in those units and read the output in the corresponding cubic or square unit. For example, entering dimensions in metres gives volume in m^3 directly.
Practical applications
Volume calculations come up whenever you need to know how much a container holds, from aquarium capacity to concrete poured into a footing. Surface area matters for material estimates: how much cardboard to make a box, how much paint to cover a wall, or how much insulation wrap is needed around a duct. The space diagonal is used to check whether a long item (a TV, a fishing rod, a piece of pipe) will fit inside a box or a truck. Packaging designers use the surface-to-volume ratio to minimise material for a given capacity, which is why cube-shaped containers are the most efficient.
Rectangular prism formulas at a glance
| Property | Formula | Notes |
|---|---|---|
| Volume | V = l x w x h | Cubic units |
| Total surface area | S = 2(lw + lh + wh) | Sum of all 6 faces |
| Lateral surface area | S_lat = 2h(l + w) | 4 side faces only |
| Top/bottom face | A = l x w | Each face identical |
| Front/back face | A = l x h | Each face identical |
| Left/right face | A = w x h | Each face identical |
| Space diagonal | d = sqrt(l^2 + w^2 + h^2) | Longest internal line |
| Number of faces | 6 | 3 pairs of rectangles |
| Number of edges | 12 | 3 groups of 4 parallel edges |
| Number of vertices | 8 | All right-angle corners |
Standard formulas where l = length, w = width, h = height.
Frequently asked questions
What is the formula for the volume of a rectangular prism?
Volume equals length times width times height: V = l x w x h. The result is in cubic units matching whatever unit you used for the dimensions. For example, if all three sides are in centimetres the volume is in cm^3.
How do you find the surface area of a rectangular prism?
Total surface area equals two times the sum of the three distinct face areas: S = 2(lw + lh + wh). There are six faces but only three unique sizes: two top/bottom faces each with area l x w, two front/back faces each with area l x h, and two left/right faces each with area w x h. Add all three pairs and multiply by two.
What is the space diagonal of a rectangular prism?
The space diagonal is the straight line connecting two opposite corners of the box, passing through the interior. It equals sqrt(l^2 + w^2 + h^2). This is the longest straight-line distance inside the prism, which tells you the maximum length object that can fit inside without bending.
What is the difference between total surface area and lateral surface area?
Total surface area includes all six faces. Lateral surface area counts only the four side faces and excludes the top and bottom. For a box with dimensions l x w x h, lateral area = 2h(l + w). This is useful when the top and bottom are open, such as calculating the wall area of a room or the label area of a cylindrical can without end caps.
Is a rectangular prism the same as a cuboid?
Yes. Rectangular prism and cuboid are two names for the same shape: a box with six rectangular faces, eight right-angle corners, and 12 edges. A cube is a special cuboid where all three dimensions are equal. In everyday language it is often simply called a box.
How is a rectangular prism different from a cube?
A cube is a special case of a rectangular prism in which all three dimensions (length, width, height) are identical. Every cube is a rectangular prism, but not every rectangular prism is a cube. The formulas are the same: for a cube with side s, V = s^3 and S = 6s^2.