What Is a Cube?
A cube is a three-dimensional solid bounded by six congruent square faces that meet at right angles. Because every face is the same square, a single measurement (the edge length $a$) describes the whole solid. A cube is the 3D version of a square, in the same way a square is a rectangle with equal sides. When a square prism has a square base and equal height, it becomes a cube, which is why a cube is a special square prism.
What Is a Cuboid?
A cuboid is a three-dimensional solid bounded by six rectangular faces, described by three measurements: length $l$, breadth $b$, and height $h$. Opposite faces are equal rectangles, and all faces meet at right angles. A cuboid is also called a rectangular prism, because it is a prism with a rectangular cross-section. Most real-world boxes - cartons, books, rooms - are cuboids.
Why Is a Dice a Cube but a Brick Only a Cuboid?
Roll a die and it looks the same from every side; stand a brick on end and it clearly does not.
That everyday contrast is the whole difference between a cube and a cuboid. Both are 3D shapes built from six flat faces, eight corners, and twelve edges - identical bookkeeping. What separates them is a single condition on the edges: a cube forces all three dimensions to be equal, and a cuboid lets them differ. Everything else, from the surface-area formula to the diagonal length, follows from that one rule.
What Is the Difference Between a Cube and a Cuboid?
A cube is a solid with six equal square faces, so its length, breadth, and height are all the same. A cuboid is a solid with six rectangular faces, so its length, breadth, and height can each be different. That is the core distinction; the table below unpacks what it means in practice.
Feature | Cube | Cuboid |
|---|---|---|
Face shape | Square | Rectangle |
Edge lengths | All 12 equal | Up to 3 different lengths |
Dimensions | $l = b = h$ | $l$, $b$, $h$ may differ |
Faces / edges / vertices | 6 / 12 / 8 | 6 / 12 / 8 |
Everyday example | Dice, Rubik's cube | Brick, matchbox, book |
Notice the counts of faces, edges, and vertices are identical. The difference lives entirely in the shape of those faces and the equality of those edges.
What Are the Formulas for a Cube and a Cuboid?
Each solid has the same kinds of formula, differing only because a cube uses one length while a cuboid uses three. In the cube column, $a$ is the edge length; in the cuboid column, $l$, $b$, $h$ are length, breadth, and height.
Quantity | Cube | Cuboid |
|---|---|---|
Volume | $a^3$ | $l \times b \times h$ |
Total surface area | $6a^2$ | $2(lb + bh + hl)$ |
Lateral surface area | $4a^2$ | $2h(l + b)$ |
Space (body) diagonal | $a\sqrt{3}$ | $\sqrt{l^2 + b^2 + h^2}$ |
Perimeter of edges | $12a$ | $4(l + b + h)$ |
Set $l = b = h = a$ in any cuboid formula and it collapses to the matching cube formula - for instance $2(lb + bh + hl)$ becomes $2(a^2 + a^2 + a^2) = 6a^2$. That collapse is the algebraic proof that a cube is just a cuboid with all sides equal.
What Do a Cube and a Cuboid Have in Common?
Despite the difference, the two solids share a surprising amount:
Same element counts - both have 6 faces, 12 edges, and 8 vertices.
Right angles everywhere - every pair of adjacent faces meets at 90°.
Opposite faces are congruent and parallel.
Both are prisms - a cube is a square prism, a cuboid a rectangular prism.
Both satisfy Euler's formula, $F - E + V = 2$: here $6 - 12 + 8 = 2$.
Is a Cube a Cuboid?
Yes - every cube is a cuboid, but not every cuboid is a cube. A cube meets the definition of a cuboid (six rectangular faces) because a square is a special rectangle. It simply adds the extra condition that all three dimensions are equal. This mirrors the 2D relationship exactly: every square is a rectangle, but only rectangles with equal sides are squares. So "cube" is the narrower, more special category sitting inside "cuboid." The formal set-inclusion is described in the cuboid reference.
Where Are Cubes and Cuboids Seen?
Both shapes are everywhere because right-angled boxes pack and stack efficiently.
Packaging. Cartons and shipping boxes are cuboids; sugar cubes and dice are cubes.
Architecture. Rooms and buildings are cuboidal; some modern designs use cube modules.
Storage. Warehouses stack cuboidal boxes to fill space with no gaps.
Games and puzzles. The Rubik's cube is the most famous cube; a domino set is a stack of thin cuboids.
The efficiency comes from the flat faces and right angles both share; the choice between cube and cuboid is just whether the three dimensions need to match.
Examples of the Difference Between Cube and Cuboid
Example 1
Find the volume of a cube with edge 5 cm.
Use $V = a^3$ with $a = 5$.
$$V = 5^3 = 125 \text{ cm}^3$$
Final answer: $125 \text{ cm}^3$.
Example 2
A cuboid measures 8 cm by 3 cm by 2 cm. A student finds its volume as $8^3 = 512 \text{ cm}^3$. What went wrong?
Wrong path. The student used the cube formula $a^3$, cubing only the length.
Why it breaks. The cube formula assumes all three dimensions are equal. A cuboid has three different dimensions, so its volume is the product of all three, not one cubed.
Correct. $V = l \times b \times h = 8 \times 3 \times 2 = 48 \text{ cm}^3$.
Final answer: $48 \text{ cm}^3$, using the cuboid formula, not the cube one.
Example 3
Find the total surface area of a cube with edge 4 cm.
Use $\text{TSA} = 6a^2$ with $a = 4$.
$$\text{TSA} = 6 \times 4^2 = 6 \times 16 = 96 \text{ cm}^2$$
Final answer: $96 \text{ cm}^2$.
Example 4
Find the total surface area of a cuboid 6 cm by 4 cm by 3 cm.
Use $\text{TSA} = 2(lb + bh + hl)$.
$$\text{TSA} = 2(6\cdot 4 + 4\cdot 3 + 3\cdot 6) = 2(24 + 12 + 18) = 2 \times 54 = 108 \text{ cm}^2$$
Final answer: $108 \text{ cm}^2$.
Example 5
Find the space diagonal of a cube with edge 3 cm.
Use the diagonal formula $a\sqrt{3}$.
$$d = 3\sqrt{3} \approx 5.20 \text{ cm}$$
Final answer: $3\sqrt{3} \approx 5.20 \text{ cm}$.
Example 6
A cuboid has volume $60 \text{ cm}^3$ with length 5 cm and breadth 4 cm. Find its height, then check whether it is a cube.
From $V = l \times b \times h$: $60 = 5 \times 4 \times h$, so $h = \dfrac{60}{20} = 3$ cm. Since $5$, $4$, and $3$ are all different, the solid is not a cube.
Final answer: $h = 3$ cm; it is a cuboid, not a cube.
Where Do Students Trip Up on Cube and Cuboid?
The most common mistake is reaching for a cube formula on a cuboid because it looks simpler. The habit that prevents it is to read the dimensions first: if a single number describes the solid, cube formulas apply; the moment two of the three dimensions differ, every formula must use $l$, $b$, and $h$ separately.
Mistake 1: Using the cube formula on a cuboid
Where it slips in: Computing a cuboid's volume or surface area.
Don't do this: Writing $V = a^3$ when the three sides are not equal.
The correct way: A cuboid needs all three dimensions: $V = l \times b \times h$ and $\text{TSA} = 2(lb + bh + hl)$. Substituting one edge into a cube formula is the single most frequent error on this topic.
Mistake 2: Assuming a cuboid cannot have any equal sides
Where it slips in: Deciding a solid can't be a cuboid because two of its faces are squares.
Don't do this: Ruling out "cuboid" the moment a square face appears.
The correct way: A cuboid may have some equal edges (a square prism is a cuboid). It becomes a cube only when all three dimensions match. Partial equality is still a cuboid.
Mistake 3: Confusing surface area with volume units
Where it slips in: Labelling an answer.
Don't do this: Reporting a volume in $\text{cm}^2$ or a surface area in $\text{cm}^3$.
The correct way: Volume is a cubic measure ($\text{cm}^3$); surface area is a square measure ($\text{cm}^2$). Checking the unit against the formula's power of the length catches the slip instantly.
Conclusion
The difference between a cube and a cuboid is face shape and edge equality: a cube has six equal squares; a cuboid has rectangles with up to three different lengths.
Both share 6 faces, 12 edges, and 8 vertices, and both satisfy Euler's formula.
Cube formulas use one edge $a$; cuboid formulas use $l$, $b$, and $h$, and setting them equal collapses cuboid formulas to cube ones.
Every cube is a cuboid, but not every cuboid is a cube.
Match the formula to the number of distinct dimensions, and keep square units for area and cubic units for volume.
To explore solid geometry with a teacher, try Bhanzu's geometry tutor or middle school math tutor, or browse math classes online.
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Practice These to Solidify Your Understanding
Work through these problems in order:
Find the volume of a cube with edge 7 cm.
Find the total surface area of a cuboid 10 cm by 5 cm by 2 cm.
A cuboid has length 6 cm, breadth 6 cm, and height 6 cm. What special solid is it, and what is its volume?
Answer to Question 1: $V = 7^3 = 343 \text{ cm}^3$. Answer to Question 2: $\text{TSA} = 2(10\cdot 5 + 5\cdot 2 + 2\cdot 10) = 2(50 + 10 + 20) = 160 \text{ cm}^2$. Answer to Question 3: All sides equal, so it is a cube; $V = 6^3 = 216 \text{ cm}^3$.
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