Cuboid Surface Area Calculator

This rectangular prism surface area calculator allows you to calculate the total surface area of a rectangular prism with six sides, also called a hexahedron. By entering length, width and height, the areas of all six sides are added together. The tool can also be used to find a missing side by inputting the surface area as well as the lengths of two known sides. During calculation, volume, space diagonal and individual faces will also be shown.

What is the surface area of a cuboid?

A cuboid is a solid shape with six flat sides, each side being a rectangle and all the angles are right angles. Shoeboxes, bricks, shipping crates, and fish tanks are examples of cuboids. As two opposite sides are identical, the six sides can be split into three pairs of congruent faces.

The surface area is the sum of the areas of all six sides, so it's the total area of material that will be required to cover the entire body, not counting overlaps. Since each side is a two-dimensional shape, the cube itself is a three-dimensional object, but its surface area is measured in square units such as square centimeters or square feet.

How to use this calculator tool:

Enter the length, width and height into the corresponding fields and select the desired unit. Once all three values have been entered, the tool will immediately display the total surface area as the main result and also provide information on volume, space diagonal and surface contribution of each side pair.

It can also be used in reverse. If the surface area and lengths of two sides are known, then the length of the third side can be calculated by entering them into a blank field. To correct it simply delete an entry field and enter a new value. When opening the chart you will see the pattern and scales for square and cube numbers.

Formula for surface area of a cuboid:

Let's call the lengths of the three sides l (length), w (width) and h (height). Since opposite sides are equal in length, we can work out the area of each of the 3 different rectangles and then double each one before adding them together.

The three products in parentheses represent the three different sides respectively. The factor 2 is used to take into account the same area on the opposite side of the box. Here, A stands for surface area and l, w, and h stand for length, width, and height respectively.

The three pairs of opposite faces:

If you split the formula into three pairs of opposite faces, it becomes clearer how the area is divided. The top and bottom each have an area that is calculated from the product of length and width, the front and back have an area that is calculated from the product of length and height, and the left and right sides have an area that is calculated from the product of width and height.

The calculator shows the sum of each of the three pairs of opposite faces and displays them in a pie chart so that it is immediately apparent which face takes up the most area. Long, flat crates will be dominated by the top and bottom faces while tall, narrow crates will be dominated by the side faces.

A full example:

Suppose we have a box with length of 5cm, width of 3cm and height of 4cm. First, we calculate the area of the three different types of faces and double each before adding them together.

Thus the surface area is 94 square centimeters. Note that the areas of the three pairs of opposite faces are 30, 40 and 24 square centimeters respectively, which is exactly the breakdown shown by the calculation tool in the results section.

The difference between a cuboid and a cube:

A cube is a box in which the length, width and height are all the same. If we call the length of each side s, then the formulas can be simplified since all six faces are identical squares with area s*s.

Thus the surface area of a cube with side length 2 cm is 24 square centimeters, which is equal to 6 times the square of 2. You can check this result using a calculator by entering the length, width, and height all as 2.

Additional measurements: volume and room diagonal

While surface area is the amount of material needed to cover the outside of a box, the calculator also shows two other related measurements that can be useful. Volume tells you how much the inside of your box can hold and is calculated by multiplying the lengths of the sides.

The space diagonal is the longest straight line inside a box, running from one corner through the center to the opposite corner. It results from an extension of the Pythagorean theorem into three dimensions.

The diagonal can help with practical problems such as determining whether a long rod or pipe will fit inside a particular wooden box.

The reason is that surface area increases in proportion to the square of the dimensions.

The surface is made up of terms that each multiply two lengths together, so it scales with the square of the dimensions. If every side of a box doubles in size, its surface not only quadruples but becomes four times as large. The volume is made up of the product of three lengths and therefore follows cubic scaling. If every side doubles in size, the volume octuples.

The square-cube law explains why larger vessels are more efficient and better insulated. As the volume of a box grows faster than its surface area as it gets bigger, there is less material per unit of volume to enclose or insulate or dissipate heat. This also explains why small animals lose heat faster than large ones, and why smaller pieces of ice melt faster than larger ones.

List of formulas for cuboids:

This table summarizes the various relationships used by a calculation tool. As a cuboid is defined by its three side lengths, these three measurements are sufficient to calculate all other properties.

Examples of surface for different boxes:

These examples show how surface area and volume change as the shape of a box changes. The first row shows a cube. All numbers are in centimeters.

Application examples for the surface of a cuboid.

Surface area is useful for finding the amount of wrapping paper or paint needed to cover an object. Packaging designers calculate how much cardboard they need to make boxes. Plasterers think of walls and ceilings as faces of a cuboid, so that they can estimate how much plaster they will use. Gift wrappers try to use as little wrapping paper as possible, while sign-makers cut panels for signs or air-conditioning ducts.

Lateral surface area refers to the four vertical sides of a rectangular prism, not including the top and bottom. It is useful when only the walls are being considered, such as calculating labels for a box or paint needed for the four walls of a room. Engineers also consider surface area in heat transfer studies because hot objects give off heat through their surfaces.

Tips for accurate results:

Measure each side along the straight sides of the cuboid and ensure that all measurements are in the same units before calculating area. Note that surface area increases proportionally to the square of the dimensions, so even small errors on one side can be magnified in the final result. When purchasing material, plan for a buffer for overlaps, seams, and waste as the geometric surface area represents only the minimum amount needed without any wiggle room.

This tool is for general educational and everyday planning purposes. For technical applications, manufacturing or other high risk activities you should verify the calculation results against standards and tolerances required by your project.