Cube Diagonal CalculatorFree online math calculator — instant results, no sign-up
Calculate the space diagonal of a cube.
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About the Cube Diagonal Calculator
The Cube Diagonal Calculator finds the space diagonal of a cube from its side length. This is the straight-line distance through the cube’s interior between two opposite vertices, not a diagonal across a single face. With the default side length of 5, the calculator returns 8.6602540378.
- Side length
- The length of one edge of the cube, expressed in any consistent length unit such as centimeters, meters, or inches.
Formula and method
d = √(a² + a² + a²) = a√3; with a = 5, d = 5√3 ≈ 8.6602540378.
- aThe cube’s side length.
- dThe cube’s space diagonal, in the same length unit as a.
Apply the Pythagorean theorem to a square face to obtain a face diagonal of a√2. Then combine that face diagonal with a perpendicular edge: d = √((a√2)² + a²) = a√3. The calculator multiplies the entered side length by √3 and displays a decimal approximation.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Side length5
Results
- Result8.6602540378
How to use the Cube Diagonal Calculator
- 1Identify the length of one edge of the cube and its unit.
- 2Enter that value in Side length, or leave the default value of 5.
- 3Run the calculation to multiply the side length by √3.
- 4Read the Result row: for the default input, it is 8.6602540378.
- 5Interpret the result in the same unit as the side length, rounding as appropriate for your measurement.
Reading your result
The Result row gives the space diagonal: the distance between opposite vertices through the cube’s interior. For Side length = 5, Result = 8.6602540378 in the same unit as the input. This is approximately 1.732 times the side length; it is neither a face diagonal nor a distance along the cube’s edges.
Limitations
The formula assumes an ideal cube with equal edge lengths and right angles between adjacent edges. A physical cube requires a positive side length; zero describes a degenerate shape, and negative lengths are not meaningful. The displayed decimal is an approximation to the exact value a√3, and its number of digits does not imply that the input measurement is equally precise. A rectangular box with unequal dimensions requires a different formula.
Common questions
What is the difference between a cube’s space diagonal and its face diagonal?+
A space diagonal connects opposite vertices through the interior and has length a√3. A face diagonal lies on one square face and has length a√2. This calculator returns the space diagonal.
Why does a side length of 5 produce 8.6602540378?+
The cube has three equal perpendicular dimensions, so the space diagonal is √(5² + 5² + 5²) = 5√3. Its decimal approximation is 8.6602540378.
Do all space diagonals of a cube have the same length?+
Yes. A cube has four space diagonals connecting its four pairs of opposite vertices, and each has length a√3.
Can I use this calculator for a rectangular box?+
Only if the box’s length, width, and height are equal. Otherwise, its space diagonal is √(length² + width² + height²), which requires all three dimensions.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.