Cuboid Calculator
Enter the length, width, and height of a cuboid (rectangular box) to instantly find its volume, surface area, and space diagonal, with a to-scale isometric diagram and step-by-step solution.
Enter all three dimensions of the box — the calculator finds the volume, surface area, and space diagonal automatically.
Cuboid Diagram (isometric view, to scale)
Step-by-Step Solution
Here's exactly how this answer was calculated, one step at a time.
- 1
Start with the three dimensions
l = 6, w = 4, h = 3
A cuboid (rectangular box) is fully defined by its length, width, and height.
- 2
Calculate the volume
V = l × w × h = 6 × 4 × 3 = 72
The volume is the space enclosed inside the box, found by multiplying all three dimensions.
- 3
Calculate the area of each pair of opposite faces
lw = 24, wh = 12, hl = 18
A cuboid has three pairs of identical rectangular faces — top/bottom, front/back, and left/right.
- 4
Calculate the total surface area
S = 2(lw + wh + hl) = 2(24 + 12 + 18) = 108
Add the three face areas together and double the sum, since each face appears twice (once on each side of the box).
- 5
Sum the squares of the three dimensions
l² + w² + h² = 36 + 16 + 9 = 61
This extends the Pythagorean theorem into three dimensions to find the diagonal.
- 6
Take the square root to find the space diagonal
d = √(l² + w² + h²) = √61 = 7.8102
The space diagonal is the longest straight line that fits inside the box, corner to corner.
✓ Final Answer: V = 72, S = 108, d = 7.8102
Free Online Cuboid Calculator
This cuboid calculator instantly finds the volume, surface area, and space diagonal of any rectangular box from its length, width, and height. Enter all three dimensions, and the calculator returns every measurement automatically, complete with a to-scale isometric diagram and a full step-by-step solution showing exactly how each formula was applied.
Whether you're a student solving mensuration problems, a teacher preparing worked examples, a warehouse manager estimating box or pallet capacity, or someone searching for a fast and accurate volume of a cuboid calculator or surface area of a rectangular box calculator, this tool covers every common cuboid calculation you're likely to need.
A cuboid — also called a rectangular box, rectangular prism, or box shape — is one of the most common three-dimensional solids in everyday life, from shipping boxes and rooms to books and bricks. Unlike a cube, a cuboid's length, width, and height can all be different values, so all three measurements are needed to fully describe it and calculate its properties.
Cuboid Formulas Used
A cuboid is defined by three independent edge lengths: length (l), width (w), and height (h). Every other measurement is derived from these three values using the following standard mensuration formulas:
- Volume: V = l × w × h (all three dimensions multiplied together)
- Surface Area: S = 2(lw + wh + hl) (the combined area of all six rectangular faces)
- Space Diagonal: d = √(l² + w² + h²) (the longest straight line inside the box, corner to corner)
- Each pair of opposite faces is identical: top/bottom (l × w), front/back (w × h), and left/right (h × l)
How to Use This Cuboid Calculator
Using this calculator takes just two simple steps. First, enter the length, width, and height of your box into the three input fields. Second, read off the results: the calculator instantly displays the volume, surface area, and space diagonal, along with a to-scale isometric diagram and a detailed breakdown of the math behind every number.
Unlike a cube, a cuboid needs all three dimensions to be fully defined, since knowing only one or two of length, width, or height isn't enough to determine the others — there are infinitely many boxes with, say, the same volume but different shapes. That's why this calculator asks for all three values directly rather than solving from a single input.
Worked Example
Suppose you're packing a shipping box with a length of 6 units, a width of 4 units, and a height of 3 units. The volume is V = 6 × 4 × 3 = 72 cubic units. The surface area is S = 2(6×4 + 4×3 + 3×6) = 2(24 + 12 + 18) = 2 × 54 = 108 square units. The space diagonal is d = √(6² + 4² + 3²) = √(36 + 16 + 9) = √61 ≈ 7.81 units — the longest straight line that would fit inside the box from one corner to the opposite corner.
This same process works for any three positive dimensions, whether you're sizing a moving box, an aquarium, a room, or a storage container.
Understanding the Cuboid's Space Diagonal
The space diagonal of a cuboid is the longest line segment that can fit entirely inside the box, running from one corner straight through the interior to the exact opposite corner. It's found by applying the Pythagorean theorem twice: first across the base to find the diagonal of the bottom rectangle (√(l² + w²)), then again using that diagonal and the height to reach the final formula, d = √(l² + w² + h²). The diagram above marks this diagonal with a dashed green line running through the body of the box, so you can see exactly which measurement it represents.
Cuboid vs. Cube: What's the Difference?
A cube is simply a special case of a cuboid where the length, width, and height are all equal. Every cube follows the same formulas as a cuboid, but because all three dimensions match, the cube's formulas simplify considerably (V = a³ instead of l×w×h, and S = 6a² instead of 2(lw+wh+hl)). If all three of your measurements are the same, you can use our Cube Calculator instead, which lets you solve from any single known value — side, volume, surface area, or diagonal.
Where Cuboid Calculations Are Used
Cuboid geometry and mensuration formulas are used constantly in everyday life and across many fields of study:
- Students and teachers use it for geometry and mensuration homework, competitive exams, and lesson planning.
- Shipping, logistics, and warehouse teams calculate cuboid volume to determine box capacity, pallet loading, and storage space requirements.
- Architects and engineers use cuboid volume and surface area when designing rooms, tanks, ducts, and rectangular structural components.
- Packaging designers calculate the surface area of a box to determine how much cardboard, wrapping, or printed material is needed.
- Interior designers and homeowners use cuboid volume to estimate paint, flooring, or the capacity of storage furniture and containers.
Frequently Asked Questions
What is the formula for the volume of a cuboid?
The volume of a cuboid is calculated using V = l × w × h, where l, w, and h are the length, width, and height. Multiply all three dimensions together to get the volume.
What is the formula for the surface area of a cuboid?
The total surface area is calculated using S = 2(lw + wh + hl), which adds up the areas of all three pairs of opposite rectangular faces and doubles the result.
How do I find the space diagonal of a cuboid?
Use the 3D version of the Pythagorean theorem: d = √(l² + w² + h²), where l, w, and h are the length, width, and height. This calculator computes it automatically once you enter all three dimensions.
What is the difference between a cuboid and a cube?
A cube is a special type of cuboid where the length, width, and height are all equal. A general cuboid can have three different dimensions, so all three values are needed to calculate its volume, surface area, and diagonal.
Why does this calculator need all three dimensions, unlike the Cube Calculator?
A cube only needs one value because all its sides are equal, but a cuboid's length, width, and height can each be different. Knowing only one or two of these values isn't enough to uniquely determine the shape, so all three are required.