Square vs Cube — Difference (with Table & FAQs)

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TL;DR: Square vs Cube compared side by side — a 7-row comparison table, 4 key points and 7 exam FAQs. Class 9-12 Maths, free and no sign-up.

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The main difference between Square and Cube: A square is a two-dimensional shape with equal sides, while a cube is a three-dimensional shape with equal edges. These geometric concepts are fundamental

Square vs Cube — Comparison Table

Basis	Square	Cube
Dimensions	Two-dimensional (2D)	Three-dimensional (3D)
Number of sides	4 sides of equal length	12 edges of equal length
Number of faces	1 flat surface	6 square faces
Vertices	4 corners	8 corners
Area/Volume	Area = side squared	Volume = side cubed
Perimeter/Surface area	Perimeter = 4 x side	Surface area = 6 x side squared
Diagonal	Side x square root of 2	Side x square root of 3

Key Points

• Square is 2D shape; cube is 3D shape made of squares
• Area measures space inside square; volume measures space inside cube
• A cube has 6 square faces; combining these faces creates the 3D shape
• Surface area of cube is sum of all 6 square faces

This question has two different answers — check which one you need

'Difference between square and cube' is asked in two quite separate senses, and answering the wrong one loses full marks. The first sense is arithmetic: squaring a number versus cubing it. The second is geometric: the square as a flat 2D shape versus the cube as a 3D solid.

Both are on the CBSE syllabus and both are covered below. If the question mentions numbers, roots or exponents, it is the arithmetic sense. If it mentions faces, edges, vertices, area or volume, it is the shape sense.

Sense 1 — squaring versus cubing a number

The square of a number is that number multiplied by itself once: n² = n × n. So 5² = 25. The cube is the number multiplied by itself twice more: n³ = n × n × n. So 5³ = 125.

The exponent is the only difference in the formula, but the growth rates diverge fast. For n = 10, the square is 100 and the cube is 1,000. For n = 20, the square is 400 and the cube is 8,000 — twenty times larger. This is why volume grows so much faster than area when an object is scaled up.

Sense 2 — the square shape versus the cube solid

A square is a two-dimensional (flat) shape with four equal sides and four right angles. It has length and breadth but no thickness, so it has area but no volume.

A cube is a three-dimensional solid built from six identical square faces. It has length, breadth and height, so it has both surface area and volume. A cube has 6 faces, 12 edges and 8 vertices. In effect, a square is one face of a cube — the cube is what you get when a square is given depth.

The formulas you need for both

Learn these as pairs; questions very often ask for one of each.

• Square of a number: n² = n × n. Square root: √n, the number which when squared gives n.
• Cube of a number: n³ = n × n × n. Cube root: ∛n, the number which when cubed gives n.
• Square shape, side a: Area = a², Perimeter = 4a, Diagonal = a√2.
• Cube solid, edge a: Volume = a³, Total surface area = 6a², Lateral surface area = 4a², Space diagonal = a√3.

Properties of square numbers that save time in exams

These let you reject wrong options instantly in an MCQ, without calculating.

• A perfect square never ends in 2, 3, 7 or 8. If a number ends in one of those digits, it cannot be a perfect square.
• A perfect square never has an odd number of zeros at the end. 100 is a perfect square; 1000 is not.
• The square of an even number is even; the square of an odd number is odd.
• Squares are never negative. Cubes can be — (−4)³ = −64, while (−4)² = +16. This sign behaviour is the most examined difference between the two operations.
• The sum of the first n odd numbers equals n². For example 1 + 3 + 5 + 7 = 16 = 4².

Perfect squares and perfect cubes to memorise

Squares 1 to 20: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.

Cubes 1 to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Notice that 1 and 64 appear in both lists — they are both perfect squares and perfect cubes (64 = 8² = 4³). The next such number is 729 = 27² = 9³.

Square vs Cube — Worked Examples

• Square and cube of 7 — 7² = 49. 7³ = 343. The cube is seven times the square, which is always true: n³ = n² × n.
• A square tile of side 5 cm — Area = 5² = 25 cm². Perimeter = 4 × 5 = 20 cm. There is no volume — the tile is treated as flat.
• A cubical box of edge 5 cm — Volume = 5³ = 125 cm³. Total surface area = 6 × 5² = 150 cm². It has 6 faces, 12 edges and 8 vertices.
• Why sign matters — (−3)² = 9, but (−3)³ = −27. Squaring destroys the negative sign; cubing keeps it.

Common Mistakes Students Make

• Answering only the number sense when the question meant the shapes, or vice versa. Read the question for the words 'faces', 'volume' or 'area'.
• Writing the total surface area of a cube as 4a². That is the lateral surface area; the total is 6a² because a cube has six faces.
• Assuming a cube is 'a square in 3D' and therefore has 4 faces. It has 6.
• Thinking a cube of a negative number is positive. Only even powers force a positive result.
• Mixing up a cube (all edges equal) with a cuboid (a box whose length, breadth and height can differ).

Diagrams

Where This Shows Up in Real Life

Why volume grows faster than area

Double the side of a cube and its surface area goes up four times but its volume goes up eight times, because area depends on a² and volume on a³. This single fact explains a great deal: why a large ice block melts more slowly than the same mass crushed into cubes (less surface per unit volume), why small animals lose body heat faster than large ones, and why a giant version of an insect could not actually support its own weight. Engineers call this the square-cube law.

Squares and cubes in everyday measurement

Whenever you buy floor tiles you are working in squares — area is measured in square metres, and the number of tiles needed depends on a². Whenever you buy water, sand or a storage tank you are working in cubes — volume is measured in cubic metres, and a tank twice as tall, wide and deep holds eight times as much. Confusing the two units, m² and m³, is the most expensive arithmetic mistake in ordinary life.

Pythagoras and squares

The theorem you use constantly in geometry is literally a statement about squares: in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, a² + b² = c². The famous 3-4-5 triangle works because 9 + 16 = 25. Recognising perfect squares instantly — 25, 144, 169 — is what lets you spot these triples in an exam without a calculator.

Cube roots in packing and design

If a cubical carton must hold 1,000 cm³, its edge is the cube root of 1,000, which is 10 cm. Working backwards from a required volume to a dimension is exactly what cube roots are for, and it is the reason the cube-root table up to 10 is worth memorising. The same reasoning sizes water tanks, shipping crates and storage boxes.

Solved Numericals — Step by Step

Board-Style Questions with Model Answers

Distinguish between the square and the cube of a number, and between a square and a cube as shapes. (3 marks)

As an operation on numbers, the square of a number multiplies it by itself once, written n² = n × n, so 5² = 25. The cube multiplies it by itself twice more, written n³ = n × n × n, so 5³ = 125. As shapes, a square is a two-dimensional figure with four equal sides and four right angles; it has length and breadth only, so it has area (a²) and perimeter (4a) but no volume. A cube is a three-dimensional solid bounded by six identical square faces, with length, breadth and height, so it has volume (a³) and total surface area (6a²), along with 6 faces, 12 edges and 8 vertices.

State four properties of perfect squares. (4 marks)

First, a perfect square never ends in the digits 2, 3, 7 or 8, which lets you rule out numbers instantly. Second, a perfect square never has an odd number of zeros at the end — 100 is a perfect square but 1000 is not. Third, the square of an even number is always even and the square of an odd number is always odd. Fourth, the sum of the first n odd natural numbers is always n²; for example 1 + 3 + 5 + 7 = 16 = 4². A fifth acceptable property is that squares are never negative, since multiplying two numbers of the same sign always gives a positive result.

A cubical tank of edge 2 m is filled with water. Find its capacity in litres and its total surface area. (3 marks)

The volume of a cube is a³, so the capacity is 2³ = 8 cubic metres. Since 1 cubic metre equals 1000 litres, the tank holds 8 × 1000 = 8000 litres. The total surface area of a cube is 6a², so it is 6 × 2² = 6 × 4 = 24 square metres. Note carefully that volume is measured in m³ and surface area in m², and that the lateral surface area would be 4a² = 16 m² if only the four walls were required.

Explain why the cube of a negative number is negative but its square is positive. (2 marks)

Squaring multiplies two identical factors together. When both are negative, their product is positive, because a negative times a negative gives a positive; so (−3)² = 9. Cubing multiplies three identical factors. The first two negatives give a positive, and multiplying that positive by the third negative factor makes the result negative; so (−3)³ = −27. In general, an even power of a negative number is positive and an odd power is negative.

Find the smallest number by which 3087 must be divided to obtain a perfect cube. (3 marks)

First find the prime factorisation. 3087 = 3 × 1029 = 3 × 3 × 343 = 3² × 7³. For a perfect cube, each prime must occur in groups of three. Here 7 occurs three times, which is complete, but 3 occurs only twice, which is an incomplete group. Dividing by 3² = 9 removes the incomplete group entirely, giving 3087 ÷ 9 = 343 = 7³, which is a perfect cube with cube root 7. The required smallest number is therefore 9.

Assertion–Reason Questions (CBSE Format)

For each question choose: (a) both A and R are true and R is the correct explanation of A; (b) both are true but R is not the correct explanation; (c) A is true but R is false; (d) A is false but R is true.

Quick Self-Check

Key Terms Glossary

Square of a number

The number multiplied by itself once: n² = n × n. So 7² = 49.

Cube of a number

The number multiplied by itself twice more: n³ = n × n × n. So 7³ = 343.

Square root

The number which, when squared, gives the original. Written √n. √49 = 7.

Cube root

The number which, when cubed, gives the original. Written ∛n. ∛343 = 7.

Perfect square

A number that is the square of a whole number, such as 1, 4, 9, 16, 25.

Perfect cube

A number that is the cube of a whole number, such as 1, 8, 27, 64, 125.

Square (shape)

A 2D figure with four equal sides and four right angles. Area a², perimeter 4a, diagonal a√2.

Cube (solid)

A 3D solid with six identical square faces. Volume a³, total surface area 6a², 12 edges, 8 vertices.

Cuboid

A box-shaped solid whose length, breadth and height may all differ. A cube is the special case where all three are equal.

Lateral surface area

The area of the side faces only, excluding the top and bottom. For a cube it is 4a².

Prime factorisation

Writing a number as a product of primes. Grouping those primes in pairs gives square roots, in threes gives cube roots.

Square-cube law

When a solid is scaled up, its area grows as the square of the scale factor but its volume as the cube.

One-Minute Revision

• TWO meanings — check whether the question is about NUMBERS or SHAPES before answering.
• Numbers: n² = n × n (one extra factor). n³ = n × n × n (two extra factors). n³ = n² × n.
• Shapes: SQUARE is 2D with area only. CUBE is 3D with volume AND surface area.
• Cube: 6 faces, 12 edges, 8 vertices. Volume a³, TOTAL surface 6a², LATERAL 4a².
• Square: area a², perimeter 4a, diagonal a√2. Cube space diagonal a√3.
• A perfect square NEVER ends in 2, 3, 7 or 8 — use this to eliminate MCQ options instantly.
• Squares are never negative. Cubes keep the sign: (−4)² = +16 but (−4)³ = −64.
• 1 and 64 are both perfect squares AND perfect cubes. So is 729 = 27² = 9³.

Syllabus reference: NCERT Class 8 Maths, Chapter 6 — Squares and Square Roots and Chapter 7 — Cubes and Cube Roots; shapes covered in Class 8 Chapter 10 — Visualising Solid Shapes.

Frequently Asked Questions

What is the difference between the square and the cube of a number?

The square of a number is that number multiplied by itself once (n × n = n²), for example 5² = 25. The cube is the number multiplied by itself twice (n × n × n = n³), for example 5³ = 125.

What is the difference between a square and a cube shape?

A square is a 2-D (flat) shape with 4 equal sides. A cube is a 3-D solid with 6 equal square faces, 12 edges and 8 vertices.

What are the square and cube of 4?

The square of 4 is 4 × 4 = 16. The cube of 4 is 4 × 4 × 4 = 64.

What is the difference between a square and a cube?

In arithmetic, the square of a number multiplies it by itself once (n² = n × n) while the cube multiplies it by itself twice more (n³ = n × n × n). In geometry, a square is a flat 2D shape with four equal sides and only area, while a cube is a 3D solid made of six square faces with both surface area and volume.

Is a cube made of squares?

Yes. A cube is bounded by six identical square faces. That is why its total surface area is 6a² — six squares, each of area a².

Can a number be both a perfect square and a perfect cube?

Yes. Such numbers are perfect sixth powers. The first few are 1, 64 (which is 8² and 4³) and 729 (which is 27² and 9³).

Why is the cube of a negative number negative?

Cubing multiplies three negative factors together. Two negatives give a positive, and the third negative makes the result negative again. Squaring uses only two factors, so the negatives always cancel and the answer is positive.

More Maths Comparisons

• Difference Between Rational and Irrational Numbers
• Difference Between Prime and Composite Numbers
• Difference Between Area and Perimeter
• Difference Between Square and Rectangle
• Difference Between Square and Rhombus
• Difference Between Mean and Median

See all difference-between comparisons →

Revising Class 9? Practise this chapter

• NCERT solutions: A Square And A Cube →
• Practice MCQs: Squares →
• Revision notes: Cubes Cube Roots →
• Previous year questions: Squares And Square Roots →