Increasing the Dimensions by a Factor of 2

Oct 13, 2025 by magicalceylontours.com 70 views

Introduction

When it comes to wrapping a rectangular prism in wrapping paper, the surface area of the prism plays a crucial role in determining the amount of paper needed. In this article, we will explore how to calculate the surface area of a rectangular prism and then apply this knowledge to find the new surface area when each dimension is increased by a factor of 2.

The Original Surface Area

To find the original surface area of the prism, we need to calculate the area of each face and then add them up. The formula for the surface area of a rectangular prism is:

SA = 2lw + 2lh + 2wh

where l, w, and h are the length, width, and height of the prism, respectively.

Given the dimensions of the prism are 4 cm by 6 cm by 3 cm, we can plug these values into the formula:

SA = 2(4)(6) + 2(4)(3) + 2(6)(3) SA = 48 + 24 + 36 SA = 108 cm^2

So, the original surface area of the prism is 108 cm^2.

Increasing the Dimensions by a Factor of 2

Now, let's say we want to increase each dimension of the prism by a factor of 2. This means the new dimensions will be:

Length (l): 4 cm x 2 = 8 cm
Width (w): 6 cm x 2 = 12 cm
Height (h): 3 cm x 2 = 6 cm

We can now use these new dimensions to find the new surface area of the prism.

The New Surface Area

Using the same formula for the surface area of a rectangular prism, we can plug in the new dimensions:

SA = 2(8)(12) + 2(8)(6) + 2(12)(6) SA = 192 + 96 + 144 SA = 432 cm^2

So, the new surface area of the prism is 432 cm^2.

Conclusion

In this article, we calculated the original surface area of a rectangular prism with dimensions 4 cm by 6 cm by 3 cm and then found the new surface area when each dimension was increased by a factor of 2. We used the formula for the surface area of a rectangular prism and plugged in the new dimensions to find the new surface area.

The Importance of Surface Area

The surface area of a rectangular prism is an important concept in mathematics and has many real-world applications. For example, in architecture, the surface area of a building is used to determine the amount of materials needed for construction. In engineering, the surface area of a machine part is used to determine the amount of heat transfer and friction.

Real-World Applications

The concept of surface area has many real-world applications, including:

Architecture: The surface area of a building is used to determine the amount of materials needed for construction.
Engineering: The surface area of a machine part is used to determine the amount of heat transfer and friction.
Packaging: The surface area of a package is used to determine the amount of wrapping paper needed.
Design: The surface area of a design is used to determine the amount of materials needed for production.

Final Thoughts

Introduction

In our previous article, we explored how to calculate the surface area of a rectangular prism and then applied this knowledge to find the new surface area when each dimension was increased by a factor of 2. In this article, we will answer some frequently asked questions related to the surface area of a rectangular prism.

Q: What is the formula for the surface area of a rectangular prism?

A: The formula for the surface area of a rectangular prism is:

SA = 2lw + 2lh + 2wh

where l, w, and h are the length, width, and height of the prism, respectively.

Q: How do I calculate the surface area of a rectangular prism with dimensions 5 cm by 7 cm by 2 cm?

A: To calculate the surface area of a rectangular prism with dimensions 5 cm by 7 cm by 2 cm, we can plug these values into the formula:

SA = 2(5)(7) + 2(5)(2) + 2(7)(2) SA = 70 + 20 + 28 SA = 118 cm^2

Q: What happens to the surface area of a rectangular prism when each dimension is increased by a factor of 3?

A: When each dimension of a rectangular prism is increased by a factor of 3, the surface area of the prism is increased by a factor of 3^2 = 9. This is because the surface area of a rectangular prism is proportional to the square of the dimensions.

Q: Can I use the same formula to calculate the surface area of a cube?

A: Yes, you can use the same formula to calculate the surface area of a cube. A cube is a special type of rectangular prism where all the dimensions are equal. For example, if a cube has a side length of 4 cm, the surface area of the cube is:

SA = 2(4)(4) + 2(4)(4) + 2(4)(4) SA = 32 + 32 + 32 SA = 96 cm^2

Q: How do I calculate the surface area of a rectangular prism with dimensions 3 cm by 4 cm by 5 cm and a hole of radius 1 cm in the center?

A: To calculate the surface area of a rectangular prism with dimensions 3 cm by 4 cm by 5 cm and a hole of radius 1 cm in the center, we need to subtract the area of the hole from the surface area of the prism. The area of the hole is:

A = πr^2 A = π(1)^2 A = π cm^2

The surface area of the prism is:

SA = 2(3)(4) + 2(3)(5) + 2(4)(5) SA = 24 + 30 + 40 SA = 94 cm^2

The surface area of the prism with the hole is:

SA = 94 cm^2 - π cm^2

Q: Can I use the same formula to calculate the surface area of a sphere?

A: No, you cannot use the same formula to calculate the surface area of a sphere. The surface area of a sphere is given by the formula:

SA = 4πr^2

where r is the radius of the sphere.

Conclusion

In this article, we answered some frequently asked questions related to the surface area of a rectangular prism. We also explored how to calculate the surface area of a cube and a sphere. By understanding how to calculate the surface area of different shapes, we can apply this knowledge to a variety of situations, from architecture to design.