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Diagonal of Cube Calculator

Geometry

Space Diagonal (d)

10.3923 m

Face Diagonal (f)

8.4853 m

Side Length (a)

6.0000 m

Surface Area

216.0000 m²

Volume

216.0000 m³

Cube Diagram

ABa = 6.00 md = 10.39 mf = 8.49 m

About This Tool

Diagonal of Cube Calculator – Space and Face Diagonal Explained

The space diagonal (also called the body diagonal) of a cube is the longest straight line that fits inside it — a line connecting two opposite vertices that passes straight through the cube's interior. This calculator finds the space diagonal and the face diagonal from a cube's side length, or reverses the process to solve for the side length when a diagonal is already known. It's useful in architecture, packaging design, 3D modeling, and geometry education whenever you need the maximum dimension that will fit inside — or pass through — a cube-shaped space.

The Space Diagonal Formula

The space diagonal is derived by applying the Pythagorean theorem twice. First, the diagonal across one face of the cube is found using the two edges that form that face:

f = √(a² + a²) = a√2

That face diagonal then becomes one leg of a second right triangle, with the cube's height as the other leg and the space diagonal as the hypotenuse:

d = √(f² + a²) = √(2a² + a²) = √(3a²) = a√3

For example, a cube with a side length of 6 cm has a face diagonal of 6 × √2 ≈ 8.4853 cm and a space diagonal of 6 × √3 ≈ 10.3923 cm. No external library is required — both formulas rely only on Math.sqrt().

Reverse Calculations

If you already know a diagonal and need the side length, rearrange either formula:

a = d / √3 — side length from the space diagonal
a = f / √2 — side length from the face diagonal

Continuing the example above, a space diagonal of 10.3923 cm gives a = 10.3923 / √3 ≈ 6 cm, and a face diagonal of 8.4853 cm gives a = 8.4853 / √2 ≈ 6 cm — both recover the original side length.

Unit Conversion

Enter the side length or diagonal in any supported unit — millimeters, centimeters, meters, kilometers, inches, feet, or yards — and the calculator converts internally to a consistent base unit before computing the result, then displays the answer back in your chosen unit.

Space diagonal vs. face diagonal
The face diagonal only crosses one flat square face of the cube, so it's always shorter than the space diagonal, which cuts through the cube's interior to the farthest possible vertex. For a unit cube, the face diagonal is √2 ≈ 1.4142 and the space diagonal is √3 ≈ 1.7321.

Surface Area and Volume

Alongside the diagonals, the calculator also reports the cube's surface area and volume for convenience:

Surface Area = 6a²
Volume = a³

Common Applications

  • Packaging and shipping — determining the longest item that can fit diagonally inside a cube-shaped box
  • Architecture and construction — checking clearances for cube-shaped rooms, elevator shafts, or storage units
  • 3D modeling and game design — computing bounding diagonals for collision detection and camera framing
  • Furniture and moving — verifying whether a large cube-shaped object can be tilted diagonally through a doorway
  • Geometry education — demonstrating how the Pythagorean theorem extends from two dimensions into three

Precision and Accuracy

All calculations use standard 64-bit floating-point arithmetic (IEEE 754), accurate to about 15 significant digits. Displayed values are rounded to your chosen precision, from 0 to 10 decimal places.

Frequently Asked Questions

Is the Diagonal of Cube Calculator free?

Yes, Diagonal of Cube Calculator is totally free :)

Can I use the Diagonal of Cube Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Diagonal of Cube Calculator?

Yes, any data related to Diagonal of Cube Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this diagonal of cube calculator work?

Enter a cube's side length to instantly get its space diagonal (the longest line through the interior, a√3) and its face diagonal (the diagonal across one face, a√2). You can also switch to reverse mode and enter either diagonal to solve for the side length, along with the cube's surface area and volume.

What is the formula for the space diagonal of a cube?

The space diagonal is d = a√3, where a is the side length. It comes from applying the Pythagorean theorem twice: first to find the face diagonal (f = a√2), then again using the face diagonal and the height to reach the opposite corner through the cube's interior.

What is the difference between the space diagonal and the face diagonal?

The face diagonal (f = a√2) runs across one flat face of the cube, connecting two corners of that square face. The space diagonal (d = a√3) is longer and passes through the interior of the cube, connecting two opposite vertices on entirely different faces.

How do I find the side length if I only know the diagonal?

If you know the space diagonal, the side length is a = d / √3. If you know the face diagonal instead, use a = f / √2. Both reverse modes are available in this calculator — just select the mode and enter the known diagonal.

Can I convert between different units?

Yes. Choose from millimeters, centimeters, meters, kilometers, inches, feet, or yards. The calculator performs all internal math in a consistent unit and displays results in your selected unit.

How accurate are the results?

Calculations use standard 64-bit floating-point arithmetic (IEEE 754), accurate to about 15 significant digits. Displayed values are rounded to your chosen precision, from 0 to 10 decimal places.