Cuboid Surface Area Calculator
Enter the length, width, and height of a cuboid (rectangular box) to get the total surface area, lateral surface area, and the area of each pair of faces. Switch between metric and imperial units and see the full working shown step by step.
Formula
Worked example
A box is 10 cm long, 7 cm wide, and 5 cm tall. Face pairs: 10x7 = 70, 10x5 = 50, 7x5 = 35. TSA = 2(70 + 50 + 35) = 2 x 155 = 310 cm2. Lateral SA = 2 x 5 x (10 + 7) = 170 cm2. Diagonal = sqrt(100 + 49 + 25) = sqrt(174) = 13.19 cm. Volume = 350 cm3.
What is a cuboid?
A cuboid (also called a rectangular prism or rectangular box) is a three-dimensional solid with six rectangular faces. Every interior angle is 90 degrees. Each pair of opposite faces is identical, which means there are three distinct face shapes: the top/bottom pair, the front/back pair, and the left/right pair. Everyday examples include bricks, shoeboxes, textbooks, suitcases, and rooms. A cube is a special cuboid where all three dimensions are equal.
The surface area formulas explained
The total surface area (TSA) adds up all six faces: TSA = 2(lw + lh + wh). Each term covers one pair of identical rectangles, and the factor of 2 accounts for both faces in the pair. The lateral surface area (LSA) covers only the four vertical sides and ignores the top and bottom: LSA = 2h(l + w). This is useful for tasks such as calculating how much paint is needed for the sides of a box, or how much material is needed for the walls of a room. The top and bottom pair is simply 2lw, the front and back pair is 2lh, and the left and right pair is 2wh.
Space diagonal and volume
The space diagonal is the longest straight line that fits inside a cuboid, running from one corner to the diagonally opposite corner. It equals sqrt(l2 + w2 + h2), which is the three-dimensional version of the Pythagorean theorem. If a cuboid can hold a rod or a long item, the space diagonal tells you the maximum length that fits without bending. Volume (l x w x h) is included alongside surface area because the two quantities often come up together, for example when comparing how much material a container uses relative to how much it holds.
Practical applications
Cuboid surface area calculations arise in packaging design, carpentry, tiling, painting, thermal insulation, sheet-metal work, and shipping logistics. For packaging, the TSA tells you how much card or plastic sheeting is needed per box, though a 5-10% offcut allowance is standard for overlaps and seams. For painting, use the lateral surface area for walls and add the top area if the ceiling is included. For wrapping paper, the TSA plus 15-20% gives a reliable estimate. For heat-loss calculations in construction, the lateral and top surface areas drive the insulation requirement.
Common cuboid dimensions and surface areas
| Object | Dimensions (cm) | Total Surface Area (cm2) |
|---|---|---|
| Standard brick | 21.5 x 10.3 x 6.5 | 1,063 |
| A4 ream of paper | 29.7 x 21.0 x 5.0 | 2,157 |
| Shoebox (average) | 33 x 20 x 12 | 3,432 |
| Carry-on suitcase | 55 x 40 x 23 | 11,290 |
| Standard moving box | 45 x 35 x 35 | 11,200 |
| Pallet (EUR-1) | 120 x 80 x 14.4 | 33,984 |
Typical cuboid objects with approximate dimensions and total surface areas.
Frequently asked questions
What is the formula for the total surface area of a cuboid?
TSA = 2(lw + lh + wh), where l is length, w is width, and h is height. Each term in the bracket is the area of one pair of identical rectangular faces. For example, a box measuring 10 cm x 7 cm x 5 cm has TSA = 2(70 + 50 + 35) = 310 cm2.
What is the difference between total and lateral surface area?
Total surface area (TSA) includes all six faces. Lateral surface area (LSA) includes only the four vertical side faces and leaves out the top and bottom. LSA = 2h(l + w). Use LSA when you only need to cover the sides, for example when painting a room without the ceiling or floor.
How is a cuboid different from a cube?
A cube is a special case of a cuboid where all three dimensions are equal (l = w = h). For a cube with side s, the surface area simplifies to 6s2 and the volume to s3. Every cube is a cuboid, but most cuboids are not cubes.
What is the space diagonal of a cuboid?
The space diagonal is the longest straight line inside the cuboid, connecting two opposite corners. Its length is sqrt(l2 + w2 + h2). This is the three-dimensional extension of the Pythagorean theorem and tells you the maximum length of an object that will fit diagonally inside the box.
How do I find a missing dimension if I know the surface area?
If you know the surface area (SA) and two dimensions, rearrange the formula. For example, to find length given SA, width, and height: l = (SA/2 - wh) / (w + h). The same algebra applies to finding width or height from the other two dimensions and the surface area.
Can I use this calculator for a room?
Yes. Enter the room length, width, and height to get the total wall, floor, and ceiling area. Use the lateral surface area for just the four walls, or subtract door and window openings as needed. If you only want the floor area, multiply length by width and double it to get the floor-plus-ceiling figure.