The Ratio of Perimeters of One Face

Oct 13, 2025 by pinkandtrash.com 105 views

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Introduction

When dealing with similar figures, we often need to find the ratios of their corresponding side lengths. In this article, we will explore how to determine the ratios for various properties of two similar cubes, given that the ratio of their side lengths is $\frac{2}{5}$ .

The Ratio of Perimeters of One Face

To find the ratio of the perimeters of one face of the smaller cube compared to the perimeter of one face of the larger cube, we need to understand that the perimeter of a cube is the sum of the lengths of its four sides.

Let's denote the side length of the smaller cube as $s_1$ and the side length of the larger cube as $s_2$ . We are given that the ratio of their side lengths is $\frac{2}{5}$ , so we can write:

\frac{s_1}{s_2} = \frac{2}{5}

Since the perimeter of one face of a cube is equal to the sum of the lengths of its four sides, we can write:

P_1 = 4s_1

P_2 = 4s_2

where $P_1$ and $P_2$ are the perimeters of one face of the smaller and larger cubes, respectively.

Now, we can find the ratio of the perimeters of one face of the smaller cube compared to the perimeter of one face of the larger cube by dividing the perimeter of the smaller cube by the perimeter of the larger cube:

\frac{P_1}{P_2} = \frac{4s_1}{4s_2} = \frac{s_1}{s_2} = \frac{2}{5}

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

The Ratio of Surface Areas

The surface area of a cube is equal to the sum of the areas of its six faces. Since the ratio of the side lengths of the two cubes is $\frac{2}{5}$ , we can find the ratio of their surface areas by squaring this ratio:

\frac{SA_1}{SA_2} = \left(\frac{s_1}{s_2}\right)^2 = \left(\frac{2}{5}\right)^2 = \frac{4}{25}

where $SA_1$ and $SA_2$ are the surface areas of the smaller and larger cubes, respectively.

The Ratio of Volumes

The volume of a cube is equal to the cube of its side length. Since the ratio of the side lengths of the two cubes is $\frac{2}{5}$ , we can find the ratio of their volumes by cubing this ratio:

\frac{V_1}{V_2} = \left(\frac{s_1}{s_2}\right)^3 = \left(\frac{2}{5}\right)^3 = \frac{8}{125}

where $V_1$ and $V_2$ are the volumes of the smaller and larger cubes, respectively.

The Ratio of Diagonal Lengths

The diagonal length of a cube can be found using the Pythagorean theorem. Since the ratio of the side lengths of the two cubes is $\frac{2}{5}$ , we can find the ratio of their diagonal lengths by taking the square root of the ratio of their side lengths squared:

\frac{d_1}{d_2} = \sqrt{\left(\frac{s_1}{s_2}\right)^2} = \sqrt{\left(\frac{2}{5}\right)^2} = \frac{2}{5}

where $d_1$ and $d_2$ are the diagonal lengths of the smaller and larger cubes, respectively.

Conclusion

In this article, we have explored how to determine the ratios for various properties of two similar cubes, given that the ratio of their side lengths is $\frac{2}{5}$ . We have found that the ratio of the perimeters of one face of the smaller cube compared to the perimeter of one face of the larger cube is also $\frac{2}{5}$ . Additionally, we have found the ratios of the surface areas, volumes, and diagonal lengths of the two cubes.

By understanding these ratios, we can better comprehend the relationships between the properties of similar figures and make more informed decisions in various mathematical and real-world applications.

References

[1] "Similar Figures" by Math Open Reference. Retrieved from https://www.mathopenref.com/similarfigures.html
[2] "Cube" by Math Is Fun. Retrieved from https://www.mathisfun.com/geometry/cube.html

Introduction

In our previous article, we explored the properties of similar cubes and how to determine the ratios of their corresponding side lengths. In this article, we will answer some frequently asked questions about similar cubes and their properties.

Q: What is the ratio of the side lengths of two similar cubes?

A: The ratio of the side lengths of two similar cubes is given by the ratio of their corresponding side lengths. For example, if the side length of the smaller cube is $s_1$ and the side length of the larger cube is $s_2$ , then the ratio of their side lengths is $\frac{s_1}{s_2}$ .

Q: How do I find the ratio of the perimeters of one face of the smaller cube compared to the perimeter of one face of the larger cube?

A: To find the ratio of the perimeters of one face of the smaller cube compared to the perimeter of one face of the larger cube, you can use the formula:

\frac{P_1}{P_2} = \frac{4s_1}{4s_2} = \frac{s_1}{s_2}

where $P_1$ and $P_2$ are the perimeters of one face of the smaller and larger cubes, respectively.

Q: What is the ratio of the surface areas of two similar cubes?

A: The ratio of the surface areas of two similar cubes is given by the square of the ratio of their corresponding side lengths. For example, if the ratio of the side lengths of the two cubes is $\frac{2}{5}$ , then the ratio of their surface areas is $\left(\frac{2}{5}\right)^2 = \frac{4}{25}$ .

Q: How do I find the ratio of the volumes of two similar cubes?

A: To find the ratio of the volumes of two similar cubes, you can use the formula:

\frac{V_1}{V_2} = \left(\frac{s_1}{s_2}\right)^3

where $V_1$ and $V_2$ are the volumes of the smaller and larger cubes, respectively.

Q: What is the ratio of the diagonal lengths of two similar cubes?

A: The ratio of the diagonal lengths of two similar cubes is given by the square root of the ratio of their corresponding side lengths squared. For example, if the ratio of the side lengths of the two cubes is $\frac{2}{5}$ , then the ratio of their diagonal lengths is $\sqrt{\left(\frac{2}{5}\right)^2} = \frac{2}{5}$ .

Q: Can I use the same formulas to find the ratios of other properties of similar cubes?

A: Yes, you can use the same formulas to find the ratios of other properties of similar cubes, such as the ratios of their surface areas, volumes, and diagonal lengths. However, you may need to modify the formulas to account for the specific property you are interested in.

Q: Are there any other properties of similar cubes that I should be aware of?

A: Yes, there are several other properties of similar cubes that you should be aware of, including:

The ratio of their surface areas to volume ratios
The ratio of their volumes to surface area ratios
The ratio of their diagonal lengths to side length ratios

These properties can be useful in a variety of mathematical and real-world applications.

Conclusion

In this article, we have answered some frequently asked questions about similar cubes and their properties. We hope that this information has been helpful in understanding the properties of similar cubes and how to determine the ratios of their corresponding side lengths.

References

[1] "Similar Figures" by Math Open Reference. Retrieved from https://www.mathopenref.com/similarfigures.html
[2] "Cube" by Math Is Fun. Retrieved from https://www.mathisfun.com/geometry/cube.html