Cuboid and Cube formulas properties and solved examples
The concept of cuboid and cube plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Mastering the differences, properties, and formulas related to these 3D shapes helps students excel in competitive exams and understand daily measurement tasks effortlessly.
What Is Cuboid and Cube?
A cuboid is a three-dimensional solid figure with six rectangular faces, twelve edges, and eight vertices, where the length, breadth, and height can all be different. A cube is a special type of cuboid where all sides are equal, so every face is a square. You’ll find this concept applied in areas such as 3D shapes, geometry, and daily-life measurements.
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 (all squares) | 12 (all equal) | 8 |
| Cuboid | 6 (rectangles) | 12 | 8 |
Primary Difference Between Cuboid and Cube
The main difference: In a cube, all edges are the same length, making every face a perfect square. In a cuboid, the edges can have different lengths, so faces are rectangles.
- Cube: All sides equal (l = b = h), all faces are squares.
- Cuboid: Sides may differ (l ≠ b ≠ h), faces are rectangles.
- Cube is a special type of cuboid where each side matches in size.
| Feature | Cube | Cuboid |
|---|---|---|
| Edge Lengths | All equal | Not all equal |
| Face Shape | Squares | Rectangles |
| Every Cube is a Cuboid? | Yes | No |
Key Formula for Cuboid and Cube
Cube:
- Surface Area: 6 × (side)2
- Volume: (side)3
- Diagonal: side × √3
Cuboid:
- Surface Area: 2(lb + bh + hl)
- Volume: l × b × h
- Diagonal: √(l2 + b2 + h2)
Step-by-Step Illustration: Example Calculation
Example 1: Find the surface area and volume of a cuboid with length 5 cm, breadth 3 cm, height 4 cm.
1. Surface Area = 2(lb + bh + hl) = 2(5×3 + 3×4 + 5×4)2. = 2(15 + 12 + 20) = 2×47 = 94 cm2
3. Volume = l × b × h = 5 × 3 × 4 = 60 cm3
Example 2: Find the volume of a cube with side 7 cm.
1. Volume = (side)3 = 73 = 343 cm3
Real-Life Usage of Cuboid and Cube
- A dice or Rubik’s cube: Cube (all sides are equal and square).
- A book, a brick, a matchbox: Cuboid (faces are rectangles, sizes vary).
- Shipping boxes: Most are cuboids, but small square gift boxes are cubes.
- Fridge, TV: Cuboids in daily life.
Classroom Tip
A quick way to remember: “If all faces are square and the box looks the same from all sides, it’s a cube. If opposite faces are rectangles of equal size but not all sides match, it’s a cuboid.” Vedantu’s teachers often use real objects in the class—like books and dice—to help students visualize these shapes.
Try These Yourself
- Find the surface area of a cube with edge 10 cm.
- Identify 3 items at home that are cuboids.
- If a cuboid has l = 6 cm, b = 2 cm, h = 3 cm, what is its volume?
- Is a refrigerator a cube or a cuboid?
Frequent Errors and Misunderstandings
- Confusing all boxes as cubes (remember: a cube has all sides equal).
- Mixing up surface area and volume formulas between cubes and cuboids.
- Using wrong measurements when sides are not labeled (always check: length, breadth, height).
Relation to Other Concepts
The idea of cuboid and cube connects with understanding squares, rectangles, and 2D–3D relationships. Mastering these shapes helps with surface area and volume topics and supports skills in advanced geometry lessons.
Cross-Disciplinary Usage
Cuboid and cube are not only useful in maths but also play an important role in physics (measuring capacity, density), computer graphics (3D modeling), and logical reasoning. Students preparing for JEE or Olympiads often encounter questions on surface areas and volumes requiring quick formula recall and comparison between these shapes.
Summary Table: Cube vs Cuboid
| Feature | Cube | Cuboid |
|---|---|---|
| Sides | Equal | Can be different |
| Faces | 6 (squares) | 6 (rectangles) |
| Diagonal | side × √3 | √(l² + b² + h²) |
| Volume | side3 | l × b × h |
Internal Links for Deeper Learning
- Difference Between Cube and Cuboid
- Surface Area and Volume of Cube, Cuboid, and Cylinder
- Faces, Edges, and Vertices
- Rectangular Prism (Cuboid)
- Properties of a Cube
We explored cuboid and cube—from meaning, formulas, and comparisons, to daily-life examples and exam tricks. Continue practicing with Vedantu to build confidence in solving all kinds of 3D geometry problems related to cubes and cuboids!
FAQs on Cuboid and Cube Concepts and Formulas Explained
1. What is a cuboid in Maths?
A cuboid is a three-dimensional solid shape with 6 rectangular faces, 12 edges, and 8 vertices. It is also called a rectangular prism. In a cuboid, opposite faces are equal and parallel, and all angles are right angles (90°). Examples of cuboids in real life include a brick, a book, and a rectangular box.
2. What is a cube?
A cube is a three-dimensional solid shape with 6 equal square faces, 12 equal edges, and 8 vertices. All edges of a cube are of equal length, and each angle is 90°. A dice or a Rubik’s cube is a common real-life example of a cube.
3. What is the formula for the volume of a cuboid?
The volume of a cuboid is calculated using the formula V = l × b × h, where l = length, b = breadth, and h = height. Volume measures the space occupied by the cuboid.
- Example: If l = 5 cm, b = 3 cm, and h = 2 cm,
- V = 5 × 3 × 2 = 30 cm³.
4. What is the formula for the volume of a cube?
The volume of a cube is given by the formula V = a³, where a is the length of one edge. Since all sides are equal, we multiply the side length three times.
- Example: If a = 4 cm,
- V = 4³ = 4 × 4 × 4 = 64 cm³.
5. What is the surface area of a cuboid?
The total surface area of a cuboid is 2(lb + bh + hl), where l = length, b = breadth, and h = height. It represents the sum of the areas of all six rectangular faces.
- Example: If l = 4 cm, b = 3 cm, h = 2 cm,
- Surface Area = 2(12 + 6 + 8) = 2(26) = 52 cm².
6. What is the surface area of a cube?
The total surface area of a cube is 6a², where a is the side length. Since all six faces are equal squares, we multiply the area of one face by 6.
- Example: If a = 5 cm,
- Surface Area = 6 × 25 = 150 cm².
7. What is the difference between a cube and a cuboid?
The main difference between a cube and a cuboid is that a cube has all edges equal, while a cuboid can have different length, breadth, and height.
- Cube: All faces are squares, all edges equal.
- Cuboid: Faces are rectangles, opposite faces equal.
- A cube is a special type of cuboid where l = b = h.
8. How many faces, edges, and vertices does a cuboid have?
A cuboid has 6 faces, 12 edges, and 8 vertices. The faces are rectangular, and each vertex is formed where three edges meet. These properties are important when identifying three-dimensional shapes in geometry.
9. How do you find the diagonal of a cuboid?
The diagonal of a cuboid is found using the formula d = √(l² + b² + h²). This formula is derived using the Pythagorean theorem in three dimensions.
- Example: If l = 3 cm, b = 4 cm, h = 12 cm,
- d = √(9 + 16 + 144) = √169 = 13 cm.
10. What are some real-life examples of cuboid and cube?
Common real-life examples of cuboid and cube help in understanding these 3D shapes in geometry.
- Cuboid: Brick, book, matchbox, rectangular box.
- Cube: Dice, ice cube, Rubik’s cube.
