The Perimeter of One Face of the Smaller Cube Compared to the Perimeter of One Face of the Larger Cube

Oct 13, 2025 by wbrum.org 21 views

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Introduction

Similar cubes are three-dimensional shapes that have the same shape but not necessarily the same size. The ratio of side lengths for two similar cubes is given as $\frac{2}{5}$ . In this article, we will determine the ratios for each of the following:

The perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube
The surface area of the smaller cube compared to the surface area of the larger cube
The volume of the smaller cube compared to the volume of the larger cube

The Perimeter of One Face of the Smaller Cube Compared to the Perimeter of One Face of the Larger Cube

To find the ratio of the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube, we need to first find the ratio of the side lengths of the smaller cube to the larger cube.

Let's assume that the side length of the smaller cube is $s$ and the side length of the larger cube is $S$ . We are given that the ratio of the side lengths is $\frac{2}{5}$ , so we can write:

\frac{s}{S} = \frac{2}{5}

Now, let's find the perimeter of one face of the smaller cube and the larger cube. The perimeter of one face of a cube is equal to the sum of the lengths of its four sides. Since all sides of a cube are equal, the perimeter of one face of the smaller cube is $4s$ and the perimeter of one face of the larger cube is $4S$ .

We can now find the ratio of the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube:

\frac{4s}{4S} = \frac{s}{S} = \frac{2}{5}

Therefore, the ratio of the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube is $\frac{2}{5}$ .

The Surface Area of the Smaller Cube Compared to the Surface Area of the Larger Cube

To find the ratio of the surface area of the smaller cube compared to the surface area of the larger cube, we need to first find the surface area of each cube.

The surface area of a cube is equal to the sum of the areas of its six faces. Since all faces of a cube are squares, the surface area of the smaller cube is $6s^2$ and the surface area of the larger cube is $6S^2$ .

We can now find the ratio of the surface area of the smaller cube compared to the surface area of the larger cube:

\frac{6s^2}{6S^2} = \left(\frac{s}{S}\right)^2 = \left(\frac{2}{5}\right)^2 = \frac{4}{25}

Therefore, the ratio of the surface area of the smaller cube compared to the surface area of the larger cube is $\frac{4}{25}$ .

The Volume of the Smaller Cube Compared to the Volume of the Larger Cube

To find the ratio of the volume of the smaller cube compared to the volume of the larger cube, we need to first find the volume of each cube.

The volume of a cube is equal to the cube of its side length. Since all sides of a cube are equal, the volume of the smaller cube is $s^3$ and the volume of the larger cube is $S^3$ .

We can now find the ratio of the volume of the smaller cube compared to the volume of the larger cube:

\frac{s^3}{S^3} = \left(\frac{s}{S}\right)^3 = \left(\frac{2}{5}\right)^3 = \frac{8}{125}

Therefore, the ratio of the volume of the smaller cube compared to the volume of the larger cube is $\frac{8}{125}$ .

Conclusion

In this article, we have determined the ratios for each of the following:

The perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube: $\frac{2}{5}$
The surface area of the smaller cube compared to the surface area of the larger cube: $\frac{4}{25}$
The volume of the smaller cube compared to the volume of the larger cube: $\frac{8}{125}$

These ratios are all based on the given ratio of side lengths for the two similar cubes, which is $\frac{2}{5}$ .

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Q: What is the definition of similar cubes?

A: Similar cubes are three-dimensional shapes that have the same shape but not necessarily the same size. They have the same ratio of corresponding side lengths.

Q: How do you find the ratio of side lengths for two similar cubes?

A: To find the ratio of side lengths for two similar cubes, you can use the formula:

\frac{s}{S} = \frac{2}{5}

where $s$ is the side length of the smaller cube and $S$ is the side length of the larger cube.

Q: What is the ratio of the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube?

A: The ratio of the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube is $\frac{2}{5}$ .

Q: What is the ratio of the surface area of the smaller cube compared to the surface area of the larger cube?

A: The ratio of the surface area of the smaller cube compared to the surface area of the larger cube is $\frac{4}{25}$ .

Q: What is the ratio of the volume of the smaller cube compared to the volume of the larger cube?

A: The ratio of the volume of the smaller cube compared to the volume of the larger cube is $\frac{8}{125}$ .

Q: How do you find the surface area of a cube?

A: The surface area of a cube is equal to the sum of the areas of its six faces. Since all faces of a cube are squares, the surface area of a cube is given by the formula:

6s^2

where $s$ is the side length of the cube.

Q: How do you find the volume of a cube?

A: The volume of a cube is equal to the cube of its side length. Since all sides of a cube are equal, the volume of a cube is given by the formula:

s^3

where $s$ is the side length of the cube.

Q: What is the difference between similar cubes and congruent cubes?

A: Similar cubes have the same shape but not necessarily the same size. Congruent cubes have the same shape and size.

Q: Can you give an example of similar cubes in real life?

A: Yes, similar cubes can be found in real life. For example, a small cube of sugar and a large cube of sugar are similar cubes. They have the same shape but not necessarily the same size.

Q: How do you use similar cubes in real life?

A: Similar cubes can be used in various applications such as architecture, engineering, and design. They can help in understanding the relationship between different shapes and sizes.

Q: What are some common applications of similar cubes?

A: Some common applications of similar cubes include:

Architecture: Similar cubes can be used to design buildings and structures.
Engineering: Similar cubes can be used to design machines and mechanisms.
Design: Similar cubes can be used to design products and packaging.

Q: Can you give an example of how similar cubes are used in architecture?

A: Yes, similar cubes can be used in architecture to design buildings and structures. For example, a small cube-shaped building and a large cube-shaped building are similar cubes. They have the same shape but not necessarily the same size.

Q: How do you find the ratio of the volume of a smaller cube compared to the volume of a larger cube?

A: To find the ratio of the volume of a smaller cube compared to the volume of a larger cube, you can use the formula:

\frac{s^3}{S^3} = \left(\frac{s}{S}\right)^3 = \left(\frac{2}{5}\right)^3 = \frac{8}{125}

where $s$ is the side length of the smaller cube and $S$ is the side length of the larger cube.