What is the volume of a cube?

V = s³, where s is the side length. For s = 5: V = 125 cubic units. Volume scales as the cube of side length — doubling side gives 8× the volume; tripling gives 27× the volume.

What is the space diagonal of a cube?

The space diagonal is d = s × √3, connecting two opposite vertices through the interior of the cube. For s = 5: d = 5√3 ≈ 8.66. Derived from Pythagorean theorem in 3D: d² = s² + s² + s² = 3s².

What is the surface area of a cube?

SA = 6 × s², because the cube has 6 identical square faces, each with area s². For s = 5: SA = 6 × 25 = 150 square units. Doubling side quadruples surface area.

How do I find side length from volume?

Take the cube root: s = ∛V. For V = 1,000: s = 10. For V = 125: s = 5. Most calculators have a cube root function (or use V^(1/3)). Unlike square root, cube root of negative numbers gives real results.

What is the face diagonal?

The diagonal across one face of the cube (a square): d_face = s√2. From Pythagorean theorem on a square face: d² = s² + s² = 2s². For s = 5: face diagonal = 5√2 ≈ 7.07. Different from space diagonal (s√3).

How many cubes can I stack in a box?

For 1 cm cubes in a 5×5×5 cm box: 5 × 5 × 5 = 125 cubes. For 2 cm cubes: only 2 fit per row, so 2 × 2 × 2 = 8 cubes. Packing problems get complex for irregular sizes. For cubes that divide evenly into box dimensions, packing is 100% efficient.

What's the relationship between cube's SA and volume?

SA/V = 6/s. Small cubes have high SA/V (high surface relative to volume). Large cubes have low SA/V. Important for: heat transfer (small things cool fast), chemistry (small particles react fast), biology (cells stay small for efficient nutrient exchange).

How is cube used in chemistry?

Many crystals have cubic structures. Rock salt (NaCl): face-centered cubic. Diamond, gold, silver: cubic close-packed. Iron: body-centered cubic. Crystal symmetry determines material properties — electronic conductivity, mechanical strength, optical behavior, etc.

Cube Calculator

Enter the side length of a cube to compute its volume, surface area, face diagonal, and space diagonal.

A cube is a three-dimensional regular polyhedron with six square faces, eight vertices, and twelve edges — the simplest and most symmetric of all 3D shapes. Cubes appear in countless contexts: dice, sugar cubes, Rubik's cubes, ice cubes, storage boxes, building blocks, shipping containers (approximately), salt crystals at microscopic scale, and pure mathematical analysis. Every face is identical; every edge equal; every angle 90°.

Cube calculations are straightforward thanks to its uniform structure. Given side length s: volume V = s³ (length × width × height, all equal). Surface area A = 6s² (six identical faces). Face diagonal d_face = s√2 (Pythagorean theorem on a square face). Space diagonal d_space = s√3 (Pythagorean theorem in 3D, connecting opposite corners through interior).

The cube is one of the five Platonic solids — regular convex polyhedra where all faces, edges, and angles are equivalent. Others are the tetrahedron (4 triangle faces), octahedron (8), dodecahedron (12 pentagons), and icosahedron (20 triangles). Among these, the cube has unique advantages: it tiles 3D space perfectly without gaps (unlike the tetrahedron).

Cubes are foundational in cryptography (cube of large numbers), computer graphics (voxels, cube mapping for environment textures), chemistry (cubic crystal structures like salt, diamond), and architecture. The classic Rubik's Cube has 43 quintillion possible states.

Common applications: packaging design (boxes are approximately cubic), construction (storage cubes, modular buildings), gaming (dice, building blocks), chemistry (crystal structures), shipping (volume calculations for cubic containers), and any 3D modeling involving cubic shapes.

Inputs

Side Length

Results

Volume

125 cubic units

Surface Area

150 sq units

Face Diagonal

7.0711

Space Diagonal

8.6603

Total Edge Length

Last updated: May 29, 2026

Formula

**Cube formulas:** Given side length s: - **Volume**: V = s³ - **Surface area**: A = 6s² - **Face diagonal** (across one face): d_f = s√2 - **Space diagonal** (corner to opposite corner): d_s = s√3 - **Total edge length**: 12s - **Perimeter of face**: 4s - **Number of faces**: 6 - **Number of edges**: 12 - **Number of vertices**: 8 **Worked example: s = 5** V = 5³ = 125 cubic units A = 6 × 25 = 150 square units Face diagonal: 5√2 ≈ 7.07 Space diagonal: 5√3 ≈ 8.66 Total edges: 60 units **Common cube sizes:** | Object | Side length | Volume | |---|---|---| | Sugar cube | 1 cm | 1 cm³ | | Dice (standard) | 1.6 cm | ~4 cm³ | | Rubik's cube | 5.7 cm | ~185 cm³ | | Ice cube (typical) | 3 cm | 27 cm³ | | Cubic foot | 30.48 cm | ~0.028 m³ | | Cubic meter | 1 m | 1 m³ | | Boring co. tunnel boring machine cross-section | ~4 m | ~64 m³ | | Shipping container (20 ft, simplified to cube) | ~2.4 m | ~14 m³ | **Volume comparisons:** | Form | Volume | |---|---| | 1 cm³ | 1 mL | | 1 dm³ | 1 L | | 1 m³ | 1,000 L = 1 kL | | 1 km³ | 10⁹ m³ = 1 trillion liters | **Inverse problems:** Given volume V, find side: s = V^(1/3) (cube root). Given surface area A: s = √(A/6). Given space diagonal d: s = d/√3. **Examples:** V = 1000 cm³ → s = 10 cm. A = 150 cm² → s = 5 cm. **Cube vs sphere (same volume):** Sphere has less surface area than cube of equal volume: - Cube: A = 6s² = 6V^(2/3) - Sphere: A = (36πV²)^(1/3) ≈ 4.836V^(2/3) Ratio: cube/sphere ≈ 1.24. Sphere is 24% more efficient for surface-to-volume ratio. Why cells, droplets, and bubbles are roughly spherical. **Surface-to-volume ratio:** For cube of side s: SA/V = 6s²/s³ = 6/s Small cubes have huge SA/V; large cubes have small SA/V. | Side | SA/V | |---|---| | 1 cm | 6 | | 5 cm | 1.2 | | 10 cm | 0.6 | | 100 cm | 0.06 | | 1 m | 6 | Important for: heat transfer (small cubes heat/cool faster), chemistry (catalyst surface area), biology (small cells exchange material faster). **Cube tilings:** A single cube perfectly tiles 3D space — no gaps, no overlaps. The only Platonic solid that does this. Used in: minerals (rock salt is cubic), packaging (efficient stacking), urban planning (grid layouts). **Cube of large numbers (volume):** 10 cm cube: 1,000 cm³ = 1 L of water. 1 m cube: 1,000 L = 1 tonne of water. 10 m cube: 1,000,000 L = 1,000 tonnes. Water-filled cube weights: useful for back-of-envelope estimates. **Cube of integers:** 1³ = 1 2³ = 8 3³ = 27 4³ = 64 5³ = 125 6³ = 216 7³ = 343 8³ = 512 9³ = 729 10³ = 1,000 **Pythagorean theorem in 3D:** Space diagonal: d² = a² + b² + c² (for general rectangular box). For cube (a = b = c = s): d² = 3s² d = s√3 **Worked example: TV size** 40-inch TV (measured diagonally) in 16:9 aspect ratio: - Width: 34.8" (88.4 cm) - Height: 19.6" (49.7 cm) - Diagonal: 40" (101.6 cm) by definition. For TV depth: thickness ~6 cm gives space diagonal still ~102 cm (negligible change). **Composites of cubes:** - **Cube root**: ∛V = side. - **Cube number**: n³ (always integer if n is integer). - **Cubical equations**: x³ = a (single real solution). **Common applications:** - **Packaging**: boxes for shipping, storage. - **Architecture**: modular buildings, cubic decorative elements. - **Game design**: Minecraft (entirely cube-based world), dice. - **Chemistry**: cubic crystal lattices (salt, diamond, gold). - **Manufacturing**: cubic storage containers, modular cubes. - **Furniture**: cubic shelving, storage cubes (IKEA Kallax). - **Construction**: precast cubic concrete blocks. **Cube symmetry:** Cube has 48 rotational symmetries (24 orientation-preserving + 24 including reflections). Highest symmetry of all real-world common solids. Used in: tiling, crystallography, group theory. **Cube nets (unfolded):** Cube has 11 distinct nets (ways to unfold into a flat connected pattern of 6 squares). Used for: cardboard box design, origami, paper craft. **Inscribed sphere vs cube:** Sphere inscribed in cube of side s: r = s/2. Sphere/cube volume ratio: (4/3 × π × (s/2)³) / s³ = π/6 ≈ 0.524. So inscribed sphere is ~52% of cube's volume. **Circumscribed sphere:** Smallest sphere containing the cube: diameter = space diagonal = s√3. Radius: s√3/2. Sphere/cube volume ratio: (4/3 × π × (s√3/2)³) / s³ = π√3/2 ≈ 2.72. Circumscribed sphere is ~2.7× the volume of cube. **Software:** - **CAD**: SolidWorks, Fusion 360 (parametric cube modeling). - **3D**: Blender (cube is default mesh). - **Game engines**: Unity, Unreal use cube primitives heavily. - **Mineralogy**: software for cubic crystal analysis.

How to use this calculator

Enter side length in your chosen units.
Calculator returns volume, surface area, face diagonal, space diagonal.
For inverse: from volume V, side = ∛V.
Units: cube side s in cm gives volume in cm³, surface in cm².
For rectangular box (not cube): use length × width × height formula instead.
For cube root: ∛n = n^(1/3).

Worked examples

Storage cube

**Scenario:** Storage cube 50 cm side. Volume, surface, diagonal? **Calculation:** V = 50³ = 125,000 cm³ = 125 L. SA = 6 × 2500 = 15,000 cm² = 1.5 m². Space diagonal: 50√3 ≈ 86.6 cm. **Result:** 125 L capacity (holds ~125 standard bottles), 1.5 m² of cardboard for outer surface, longest interior diagonal 86.6 cm (~34"). Useful for: shipping cost (volumetric weight), wrapping paper amount, fitting long items diagonally.

Rubik's cube specs

**Scenario:** Standard 3x3x3 Rubik's cube: 5.7 cm side. Total surface area? **Calculation:** SA = 6 × 5.7² = 6 × 32.49 = 194.94 cm². Per face: 32.49 cm². Per small square: 32.49/9 = 3.61 cm² (1.9 × 1.9 cm small squares). **Result:** ~195 cm² of total surface, 9 colored squares per face. Volume = 5.7³ = 185.2 cm³. 43,252,003,274,489,856,000 possible configurations (43 quintillion). World record solve: under 4 seconds.

Shipping container volume

**Scenario:** Treat 40-ft shipping container as a cube of effective side. Standard interior: 12 m × 2.35 m × 2.4 m = 67.7 m³. Equivalent cube side? **Calculation:** s = ∛67.7 ≈ 4.08 m. **Result:** ~4 m equivalent cube. Actual container is rectangular (length much greater than width/height) — cube approximation only for total volume comparison. Real packing efficiency depends on cargo shape. Cubic measure essential for shipping cost calculation ("dim weight").

When to use this calculator

**Use cube calculations for:**

- **Packaging**: box volume, surface area, materials. - **Construction**: cubic blocks, concrete, modular elements. - **Storage**: shelves, bins, organization. - **Gaming**: dice (cube), building blocks. - **Chemistry**: cubic crystals (salt, diamond). - **Logistics**: shipping volume calculations. - **Architecture**: cubic decorative elements, glass cubes. - **Education**: introducing 3D geometry to students.

**Cube vs rectangular box:**

If length ≠ width ≠ height, it's a rectangular box (rectangular prism), not a cube.

Use: V = l × w × h, SA = 2(lw + lh + wh).

Cube is a special case where l = w = h = s.

**Practical cube applications:**

- **Sugar cubes**: ~1 cm side, ~1 g sugar. - **Ice cubes**: variable sizes (2-3 cm typical). - **Bouillon cubes**: standardized small cubes. - **Storage cubes** (IKEA Kallax etc.): typically 33 or 39 cm cubes. - **Building blocks** (LEGO, Mega Blocks): rectangular but cube-related. - **Building cubes** (CMU concrete blocks): rectangular but standardized.

**Surface-to-volume ratio:**

SA/V = 6/s (for cube of side s).

Important when: - **Heat transfer**: small cubes (high SA/V) heat/cool faster. - **Chemistry**: small particles react faster. - **Biology**: cells use SA/V for efficient material exchange. - **Drying**: small cubes dry faster than large.

**Cube root:**

s = ∛V

For V = 1000: s = 10. For V = 27: s = 3. For V = 1,000,000: s = 100.

**Powers of integers (memorize):**

1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1,000.

11³ = 1,331, 12³ = 1,728, 13³ = 2,197.

**Volume conversions:**

| From | To | Multiply by | |---|---|---| | cm³ | mL | 1 | | cm³ | L | 0.001 | | dm³ | L | 1 | | m³ | L | 1,000 | | m³ | ft³ | 35.31 | | ft³ | cm³ | 28,317 | | in³ | cm³ | 16.39 | | gallon (US) | L | 3.785 | | gallon (US) | ft³ | 0.134 |

**Common applications:**

- **Shipping**: cargo volume in m³, ft³, or "cube" (industry term). - **HVAC**: room volume for air handling. - **Inventory**: cubic feet of storage space. - **Construction**: concrete volumes (m³ or yd³). - **Aquariums**: water volume (in cube-shape tanks). - **Refrigeration**: BTU/m³ cooling capacity. - **Lighting**: lumens per cube of space.

**Cubic numbers in math:**

- Cube numbers: 1, 8, 27, 64, 125, 216, ... - Sum of first n cubes: (1+2+...+n)² (beautiful identity). - Cube root: ∛x. - Pierre de Fermat's Last Theorem: no positive integer solutions to a³ + b³ = c³ (proven by Wiles).

**Software:**

- **3D CAD**: any package has cube/box primitives. - **3D printing**: cubes are simplest, often used for test prints. - **Minecraft**: built entirely from cubes (1 m^3 each in-game). - **Voxel rendering**: cube-based 3D graphics.

**Pitfalls:**

- **Confusing cube with rectangular box**: cube has all sides equal. - **Cube vs square**: square is 2D (s² area), cube is 3D (s³ volume). - **Volume vs surface area**: volume is cubed, surface is squared. - **Space diagonal vs face diagonal**: different formulas. - **Cubic units**: cm³, m³ — not cm² or m². - **Cube root of negative**: real result exists (∛-8 = -2), unlike square root. - **For irregular volume**: don't approximate as cube without thinking.

Common mistakes to avoid

Confusing cube (3D) with square (2D).
Computing surface area as s³ instead of 6s².
Confusing face diagonal (s√2) with space diagonal (s√3).
Treating rectangular box as cube (all sides must be equal for cube).
Forgetting units (cm gives cm³, not cm²).
Computing volume of cube and surface in wrong units.
Confusing cube root with square root.
For cube of side 0 or negative: nonsense answers.

Frequently Asked Questions

Sources & further reading

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