In the realm of geometry, the cube and cuboid are two important three-dimensional shapes. They are widely encountered in various fields, including architecture, engineering, and mathematics. Understanding their definitions, formulas, and properties is crucial for accurately analyzing and working with these shapes. In this article, we will explore the concepts of cube and cuboid, delve into their formulas for measurement and calculation, and examine their fundamental properties.

**Cube and Cuboid Definition**

**Cube Definition:** A cube is a three-dimensional geometric shape that has six congruent square faces, twelve edges, and eight vertices. All angles in a cube are right angles (90 degrees). The length of each side of a cube is equal, making it a special case of a cuboid with all sides of equal length.

**Cuboid Definition:** A cuboid, also known as a rectangular prism, is a three-dimensional shape with six rectangular faces, twelve edges, and eight vertices. Unlike a cube, a cuboid does not require all sides to be equal in length.

**Properties of a Cuboid:**

- A cuboid has six rectangular faces.
- The opposite faces of a cuboid are congruent and parallel.
- All angles in a cuboid are right angles (90 degrees).
- The edges of a cuboid can have different lengths.
- The diagonals of a cuboid are not necessarily equal in length and do not necessarily intersect at right angles.
- The surface area of a cuboid is the sum of the areas of its six faces.
- The volume of a cuboid is determined by multiplying the length, width, and height.

**Properties of a Cube:**

- All faces of a cube are congruent squares.
- All angles in a cube are right angles (90 degrees).
- All edges of a cube are congruent.
- The diagonals of a cube are equal in length and intersect at right angles.
- The surface area of a cube is equal to six times the area of one face.
- The volume of a cube is determined by cubing the length of one side.

## Cube and Cuboid Formulas

Shape |
Surface Area |
Volume |
---|---|---|

Cube | SA = 6 * (side length)^{2} |
V = (side length)^{3} |

Cuboid | SA = 2 * (length * width + width * height + height * length) | V = length * width * height |

### Real-life examples of cuboid

Books, match-box and pencil box, etc.

## Example of Cube and Cuboid Shape

**Problem 1: Cube Surface Area**

Find the surface area of a cube with a side length of 5 cm.

**Solution:**

Using the formula for the surface area of a cube, we have:

SA = 6 * (side length)^2

= 6 * (5 cm)^2

SA= 6 * 25 cm^2

= 150 cm^2

Therefore, the surface area of the cube is 150 square centimeters.

**Problem 2: Cube Volume**

Determine the volume of a cube with a side length of 8 meters.

**Solution:**

Using the formula for the volume of a cube, we have:

V = (side length)^3

= (8 m)^3

volume= 8 * 8 * 8 m^3

= 512 m^3

Hence, the volume of the cube is 512 cubic meters.

**Problem 3: Cuboid Surface Area**

Calculate the surface area of a cuboid with dimensions: length = 12 cm, width = 6 cm, height = 4 cm.

**Solution:**

Using the formula for the surface area of a cuboid, we have:

SA = 2 * (length * width + width * height + height * length)

= 2 * (12 cm * 6 cm + 6 cm * 4 cm + 4 cm * 12 cm)

surface area = 2 * (72 cm^2 + 24 cm^2 + 48 cm^2)

= 2 * 144 cm^2

= 288 cm^2

Therefore, the surface area of the cuboid is 288 square centimeters.

**Problem 4: Cuboid Volume**

Find the volume of a cuboid with dimensions: length = 10 meters, width = 5 meters, height = 3 meters.

**Solution:**

Using the formula for the volume of a cuboid, we have:

V = length * width * height

= 10 m * 5 m * 3 m

= 150 m^3

Thus, the volume of the cuboid is 150 cubic meters.

**Problem 5: Find the cube root of 46656 by prime factorisation method**

To find the cube root of 46656 using the prime factorization method, we need to express the number 46656 as a product of prime factors and then extract the cube root.

Let’s start by finding the prime factorization of 46656:

Dividing 46656 by 2 repeatedly, we get:

46656 ÷ 2 = 23328 (remainder 0)

23328 ÷ 2 = 11664 (remainder 0)

11664 ÷ 2 = 5832 (remainder 0)

5832 ÷ 2 = 2916 (remainder 0)

2916 ÷ 2 = 1458 (remainder 0)

1458 ÷ 2 = 729 (remainder 0)

Now, 729 is not divisible by 2, so we move on to the next prime number, which is 3:

729 ÷ 3 = 243 (remainder 0)

243 ÷ 3 = 81 (remainder 0)

81 ÷ 3 = 27 (remainder 0)

27 ÷ 3 = 9 (remainder 0)

9 ÷ 3 = 3 (remainder 0)

3 ÷ 3 = 1 (remainder 0)

At this point, we have expressed 46656 as the product of its prime factors: 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3.

Now, to find the cube root, we group the prime factors in sets of three:

(2 × 2 × 2) × (2 × 2 × 3) × (3 × 3 × 3) × (3 × 3 × 3)

Taking one factor from each group, we get:

2 × 2 × 3 × 3 = 36

Therefore, the cube root of 46656 is 36.

## Frequently Asked Questions on Cuboid and Cube

**1. What is the difference between a cube and a cuboid?**

A cube is a three-dimensional shape with six congruent square faces, while a cuboid is a shape with six rectangular faces. In a cube, all sides are equal in length, whereas a cuboid can have different side lengths.

**2. How many edges does a cube have?**

A cube has 12 edges. Each edge connects two vertices of the cube.

**3. What is the difference between the surface area and volume of a shape?**

The surface area of a shape refers to the total area covered by its outer surface. It is measured in square units. On the other hand, the volume of a shape represents the amount of space enclosed by the shape. It is measured in cubic units.

**4. Are the diagonals of a cube and a cuboid equal in length?**

In a cube, all diagonals are equal in length, and they intersect at right angles. In a cuboid, the diagonals are not necessarily equal, and their lengths depend on the dimensions of the cuboid.

**5. Can a cuboid have all sides of equal length?**

No, a cuboid cannot have all sides of equal length. A cuboid with all sides of equal length is actually a special case called a cube.

**6. How can I find the length of a side given the volume of a cube or cuboid?**

To find the length of a side, you can calculate the cube root of the volume. For example, if you know the volume of a cube is 64 cubic units, the length of each side would be the cube root of 64, which is 4 units.

**7. Can a cube have a surface area of 0?**

No, a cube cannot have a surface area of 0. Since a cube has six faces, each with a positive area, the total surface area of a cube is always greater than 0.

**8. Are all angles in a cube and cuboid right angles?**

Yes, in both a cube and a cuboid, all angles are right angles (90 degrees).

**9. Can a cuboid have all faces congruent?**

No, a cuboid cannot have all faces congruent. Congruent faces would imply that all side lengths are equal, which would make the shape a cube.

**10. What are some real-life examples of cubes and cuboids?**

Examples of cubes include dice, Rubik’s cubes, and small sugar cubes. Cuboids can be seen in objects like rectangular boxes, books, and shoeboxes.

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