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Geometry

How To Calculate The Surface Area Of A Cube

A cube is the one solid where every face is identical, so the total surface collapses to a single multiplication once you know the edge.

Quick Answer

surface area = 6 x s squared, where s is the edge length

s
Edge length
A
Total surface area

Square the edge length, multiply by six, and you have the total surface. For a 5 cm cube that is 6 x 25 = 150 square centimetres.

What Is The Surface Area Of A Cube?

The surface area of a cube is the total area of the six identical square faces that bound it. Because a square of side s has area s squared, and a cube has exactly six such faces, the whole formula is six times s squared. That is the entire calculation, and it is one of the few solid-geometry formulas that needs no approximation, no special constant and no case analysis.

Knowing the total surface area is what tells you how much material it takes to build or wrap the shape. A gift box needs six faces worth of card. A painted steel cube needs enough paint for six faces worth of coating. A heat sink machined into a cubic block exposes six faces worth of area to the surrounding air, which is why the number shows up in thermal work as often as in packaging work.

The lateral area is a separate figure that comes up whenever the cube sits on something and you do not want to cover the face in contact with the ground. The four faces standing up around the cube add to four times s squared, so the lateral area is exactly two thirds of the total. If a problem mentions a cube resting on a table, on a floor or on a base, read carefully to see whether it wants four faces or all six.

The relationship between surface area and volume is worth holding onto, because it is the reason small objects behave so differently from large ones. Surface area grows with s squared while volume grows with s cubed, so the ratio of surface to volume falls as the cube gets bigger. A 1 cm cube has 6 square centimetres of surface for each cubic centimetre of volume, which is a ratio of 6. A 10 cm cube has 0.6. Ice cubes melt fast and glaciers melt slowly for exactly this reason.

The space diagonal is the distance from one corner of the cube straight through the interior to the opposite corner, and it works out to the edge times the square root of three. It is not part of the surface area, but it is the number you need when you are checking whether a cube will fit inside a box, a crate or a shipping container. A 5 cm cube needs about 8.66 cm of clear length in every direction if it is to rotate freely inside a case.

One practical note on units: surface area is always in square units of whatever length you measured. If you measure the edge in centimetres the answer is in square centimetres, and converting to square metres requires dividing by 10,000 rather than by 100. Mixing that up is the single most common error in surface area work.

Formula

A = 6 x s^2

Six identical square faces, each with area s squared.

SymbolMeaning
ASurface area
sEdge length

A lateral = 4 x s^2

Only the four upright faces, leaving the top and bottom uncovered.

SymbolMeaning
A lateralLateral area
sEdge length

s = sqrt(A / 6)

Rearrange the surface area formula when you know the area and need the edge.

SymbolMeaning
sEdge length
ASurface area

How To Calculate The Surface Area Of A Cube

  1. 1

    Measure the edge

    Take the length of any single edge. On a real cube all twelve edges are equal, so one measurement is enough; if two disagree, the shape is a cuboid and this formula does not apply.

  2. 2

    Square the edge

    Multiply the edge length by itself to get the area of one face. A 5 cm edge gives 25 square centimetres for that face.

  3. 3

    Multiply by six

    A cube has six faces, so multiply the single face area by six. Six times 25 is 150 square centimetres.

  4. 4

    Check the units

    The result is in square units. If the edge was measured in centimetres, the answer is square centimetres; convert to square metres only by dividing by 10,000.

Examples

Example 1: A cube with 5 cm edges

Edge length
5 cm
StepCalculationResult
One faces^2 = 5 x 525 square cm
All six facesA = 6 x s^26 x 25
Total surface areaA150 square cm

Result: A cube with 5 cm edges has a total surface area of 150 square centimetres, of which 100 square centimetres is the four upright sides.

Example 2: Wrapping a 10 cm box

Edge length
10 cm
StepCalculationResult
One faces^2 = 10 x 10100 square cm
All six facesA = 6 x 100600 square cm
Paper to buyarea to buy600 square cm

Result: You need 600 square centimetres of wrapping paper, which is a 40 cm by 15 cm rectangle, before allowing anything for overlap or folds.

Example 3: Recovering the edge from a known area

Total surface area
294 square inches
StepCalculationResult
Area of one faceA / 649 square inches
Edge lengths = sqrt(A / 6)7 inches
VolumeV = s^3343 cubic inches

Result: The edge is 7 inches, and the cube holds 343 cubic inches.

Calculator

Total surface area

150

Lateral area (4 sides)
100
Volume
125
Space diagonal
8.6603

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

Prefer a full-width tool? Open the The Surface Area Of A Cube calculator page.

Common Mistakes

  • Using s squared instead of six times s squared

    The most frequent error is reporting the area of a single face as the total. Six times s squared is the total; s squared is one face out of six.

  • Counting five faces

    A cube resting on a table still has six faces and still has the surface area that follows from them. Only switch to four times s squared when the question explicitly says the base is not covered.

  • Confusing surface area with volume

    Volume is s cubed and answers how much fits inside; surface area is six s squared and answers how much wraps the outside. They are different formulas with different units.

  • Forgetting the square on the units

    Length in centimetres gives area in square centimetres. Writing the answer in centimetres is a dimensional error even when the number is right.

  • Converting area by the length factor

    One metre is 100 centimetres, but one square metre is 10,000 square centimetres. Divide or multiply by the square of the linear factor, never by the linear factor itself.

  • Applying the cube formula to a cuboid

    If the edges are not all equal, use two times lw plus two times lh plus two times wh instead. The cube shortcut only works when l, w and h are the same.

FAQ

How do you calculate the surface area of a cube?

Square the edge length and multiply by six. If the edge is 4 cm, one face is 16 square cm and all six faces total 96 square cm.

What is the lateral surface area of a cube?

It is four times the edge squared, the area of the four upright faces only. For a cube with 3 cm edges that is 36 square cm.

How do you find the edge length when you only know the surface area?

Divide the surface area by six to get one face, then take the square root. A surface area of 54 square cm gives a face of 9 square cm and an edge of 3 cm.

Is the surface area of a cube ever the same as its volume?

Only numerically, and only for an edge of 6 units, where both come out as 216. They are still different quantities with different units, so the coincidence is not physically meaningful.

Why does a cube have exactly six faces?

A cube is a regular hexahedron: six square faces, eight vertices and twelve equal edges. That count is fixed by the definition of the shape.

Does a cube have a largest face?

No. All six faces are congruent squares with identical area, which is precisely what makes the surface area formula a single multiplication.

References

  1. [1]Wikipedia, Cube surface area — https://en.wikipedia.org/wiki/Cube
  2. [2]Khan Academy, Surface area of prisms and solids — https://www.khanacademy.org/math/geometry
  3. [3]Math is Fun, Geometry: area and volume formulas — https://www.mathsisfun.com/geometry/