Calculating the Surface Area of a Cube with the Same Volume as a Cuboid

In this article, we will explore the process of determining the total surface area of a closed cube that has the same volume as a given cuboid. We will provide a detailed step-by-step solution to understand the underlying mathematical principles involved.

Understanding the Problem

The first step in solving this problem is to understand the relationship between the volume of a cuboid and the volume of a cube. A cuboid is a three-dimensional figure with six rectangular faces, and its volume can be calculated by multiplying its length, breadth, and height. A cube, on the other hand, is a three-dimensional figure with six identical square faces, and its volume can be expressed as the cube of the side length.

Volume of the Cuboid

The volume of the cuboid is given as 8 cm by 12 cm by 12 cm. The formula for the volume of a cuboid is:

V l times w times h

Substituting the given dimensions:

V 8 text{ cm} times 12 text{ cm} times 12 text{ cm} 1152 text{ cm}^3

Side Length of the Cube

To find the side length of a cube with the same volume, we need to determine the side length 'a' of the cube. The volume of a cube is given by:

V a^3

Setting the volumes equal gives us:

a^3 1152

To find the side length 'a', we take the cube root:

a sqrt[3]{1152}

Calculating the cube root of 1152:

a approx 10.26 text{ cm}

Total Surface Area of the Cube

The total surface area of a cube is given by:

A 6a^2

Substituting the side length 'a' into the surface area formula:

A 6 times (10.26 text{ cm})^2

Calculating the square of the side length:

10.26^2 approx 105.27 text{ cm}^2

Substituting back into the surface area formula:

A approx 6 times 105.27 approx 631.62 text{ cm}^2

Thus, the total surface area of the closed cube is approximately 631.62 cm2.

Mathematical Analysis and Steps

To generalize the solution, we can use the following equations and steps:

1. **Volume of the cuboid

V 8 text{ cm} times 12 text{ cm} times 12 text{ cm} 1152 text{ cm}^3

2. **Side length of the cube

a sqrt[3]{1152} approx 10.26 text{ cm}

3. **Total surface area of the cube

A 6a^2 6 times (10.26 text{ cm})^2 approx 631.62 text{ cm}^2

Conclusion

In this article, we explored the concept of finding the surface area of a cube that has the same volume as a given cuboid. By understanding the volume of the cuboid and then finding the side length of the cube, we were able to calculate the total surface area. This method can be applied to any scenario where the volume of a cuboid needs to be matched to the volume of a cube, thereby determining the surface area of the cube.

For those interested in further mathematical explorations or practical applications, consider investigating the relationship between different geometric shapes and their properties, such as volumes and surface areas.