Cube

Introduction to Cube

A cube is a solid 3D shape that has all sides equal. It has six square faces, twelve edges, and eight corners. Every face of a cube is a perfect square, and all its edges are of the same length. That is why a cube looks the same from every side.

In real life, we can see many cubes around us. Examples include a dice, an ice cube, a Rubik's cube, or a gift box. These objects are shaped like a cube because all their sides are equal and square in shape.

In mathematics, a cube is important because it allows us to calculate its surface area and volume. Learning about cubes helps us understand the 3D shapes in geometry and also connects to real-world measurements.

Table of Contents  

• Cube Definition
• Cube shape
• Area and Volume
• Properties of a Cube
• Difference Between Square and Cube
• Real-World Applications
• Solved Examples
• Practice Questions
• FAQs on Cube
  
  

Cube Definition

A cube is a 3D solid shape, has 6 square faces, 12 edges, and 8 corners. All sides are the same length, and each face meets another at a right angle. In simple words, a cube looks like a box or dice, where each side is the same.

Cube shape

A shape is the form of an object. The shapes can be flat, like a square, circle, or rectangle, or solid (3D), such as a cube, ball, or cylinder. A cube is a 3D solid shape that looks like a box or block. In a cube, length, width, and height are all equal. It has 6 square faces, 12 edges, and 8 corners. 3 edges are found in each corner. You can see examples of cubes in daily life, such as dice, ice cubes, or a gift box. The face of a cube has squares, and they are all equal in size. Because of this, a cube is the same from all sides.

Since the cube is a 3D shape, we can measure two important things:

• Surface area → Total area of ​​all 6 square faces. Formula:Surface Area = 6a²

• Volume → space inside the cube. Formula: Volume = a³ 

Area and Volume

• The area is space in a flat shape (measured in square units, like cm²).

• The volume is the shape inside a solid (cubic units, like cm³).

Example Table of Area and Volume:

 

Properties of a Cube

Regular structure: Every side, edge, and angle is congruent.

Symmetry:

• 9 symmetry planes (three planes cutting through the centres of opposite faces and six diagonal planes).

• Rotational symmetry of order 24: it can rotate about its centre in various ways to show an identical shape.  

• Face diagonals (lines across a square face) are equal.  

• Space diagonals (lines connecting opposite corners through the interior) are equal.  

• The centroid (geometric centre) is equidistant from all vertices.  

• The properties of a cube make it a preferred shape for modelling solids and creating packaging.

  

Difference Between Square and Cube

• A square is the result of multiplying a number by itself once.
  Example: 4² = 4 × 4 = 16

• A cube is the result of multiplying a number by itself twice more.
  Example: 5³ = 5 × 5 × 5 = 125

The notation used is:

• Square: n²

• Cube: n³

Squares and Cubes Chart (1 to 50)

 

Real-World Applications

• Packaging: Efficient cubic boxes with calculated volume and material.  

• Construction and Interior Design: Modelling rooms and art installations.  

• Gaming: Dice are perfect cubes with markings on faces.  

• 3D Graphics and Modelling: Voxels in games like Minecraft and 3D scanning.  

• Architecture: Floor plans subdivided into cubic elements for planning.  

• Robotics: Cubic frames used in structural scaffolding.

 

Solved Examples

1. Surface Area for a = 7 cm

Formula:
Surface Area = 6 × a²

Steps:
a = 7 cm
a² = 49
Surface Area = 6 × 49 = 294 cm²

Answer:
Surface area = 294 cm²

 

2. Volume when a = 3 in

Formula:
Volume = a³

Steps:
a = 3 in
3³ = 27

Answer:
Volume = 27 in³

 

3. Face Diagonal for a = 10 m

Formula:
Face diagonal d = a × √2

Steps:
a = 10 m
d= 10 × √2 ≈ 10 × 1.414 ≈ 14.14 m

Answer:
Face diagonal ≈ 14.14 m

 

4. Space Diagonal for a = 4 ft

Formula:
Space diagonal (d) = a × √3

Steps:
a = 4 ft
d = 4 × √3 ≈ 4 × 1.732 ≈ 6.93 ft

Answer:
Space diagonal ≈ 6.93 ft

 

5. Cubeshaped Room with Edge = 5 m

a = 5 m

Surface Area Formula:
6 × a² = 6 × 25 = 150 m²
Surface Area = 150 m²

Volume Formula:
a³ = 5³ = 125 m³
Volume = 125 m³

Face Diagonal Formula:
d_f = a × √2 = 5 × √2 ≈ 5 × 1.414 = 7.07 m
Face diagonal ≈ 7.07 m

Space Diagonal Formula:
d_s = a × √3 = 5 × √3 ≈ 5 × 1.732 = 8.66 m
Space diagonal ≈ 8.66 m

 

Practice Questions

1. What is the cube 1 to 20?

2. What is called a cube?

3. How many faces of a cube?

4. Is 64 a square and a cube number?

5. What is the cube of 40?

 

FAQs on Cube

1. What is a Cube and a Cuboid?

Answer: A cube is a 3D shape with six equal square faces, twelve equal edges, and all angles measuring 90 degrees. All sides of a cube are of the same length.

A cuboid is also a 3D shape, but has rectangular faces. Its opposite faces are equal, and it has different dimensions for length, breadth, and height. All angles are 90 degrees.

2. What are 5 Examples of Cuboids?

Answer: Here are five common examples of cuboids:

• Brick

• Matchbox

• Book

• Chocolate bar

• Shoebox

3. What is a 4-Dimensional Cuboid?

Answer: A 4-dimensional cuboid is called a tesseract.

• It extends the concept of a cube into the fourth dimension.

• A tesseract has 8 cubical cells, 24 square faces, 32 edges, and 16 vertices.

• It's used in theoretical mathematics and physics.

4. How to Find the Area of 4 Walls of a Cuboid?

Answer: The formula to calculate the area of the 4 walls (lateral surface area) of a cuboid is:

Area of 4 walls = 2 × (l + b) × h

Where:
l = length
b = breadth
h = height

Example:
If l = 6 m, b = 4 m, and h = 5 m:
Area = 2 × (6 + 4) × 5 = 2 × 10 × 5 = 100 m²

a3+b3=(a+b)(a2−ab+b2)

Explore more about Cube and practice other math concepts with our free learning resources at Orchids The International School.