Calculating Surface Area Of Similar Solids A Step-by-Step Guide

Jul 15, 2025 by ADMIN 64 views

The Volumes of Two Similar Solids and Surface Area Calculation

In the realm of geometry, understanding the relationship between similar solids is crucial. Similar solids are three-dimensional figures that have the same shape but different sizes. This means that their corresponding dimensions are proportional. When dealing with similar solids, there are important relationships between their volumes and surface areas. Specifically, if two solids are similar, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions, and the ratio of their surface areas is the square of that same ratio. This article explores how to utilize these principles to solve a practical problem: determining the surface area of a larger solid given the volumes of two similar solids and the surface area of the smaller one. We will delve into the step-by-step process, ensuring a clear and comprehensive understanding of the mathematical concepts involved.

Let's consider the following problem: We have two similar solids. The volume of the larger solid is 729 cubic inches, and the volume of the smaller solid is 125 cubic inches. The surface area of the smaller solid is 74.32 square inches. Our task is to find the surface area of the larger solid, rounding the answer to the nearest hundredth. This problem perfectly illustrates the application of similarity principles in three-dimensional geometry. By carefully analyzing the given information and applying the relevant formulas, we can accurately determine the unknown surface area. The problem requires us to understand and utilize the relationships between volumes and surface areas of similar solids, making it a valuable exercise in geometric problem-solving.

Before we dive into the solution, it's crucial to understand the concept of similarity ratios. When two solids are similar, their corresponding lengths, widths, and heights are proportional. This proportionality extends to other measurements as well, such as surface areas and volumes. The ratio of corresponding linear dimensions (like the sides) is called the scale factor. If the scale factor between two similar solids is k, then the ratio of their surface areas is k $^2$ , and the ratio of their volumes is k $^3$ . These relationships are fundamental to solving problems involving similar solids. The scale factor is the key to unlocking the connection between the dimensions, surface areas, and volumes of similar figures. A clear grasp of these ratios allows us to move seamlessly between different measurements and solve complex geometric problems with ease and accuracy. The ability to calculate and apply the scale factor is a cornerstone of understanding similarity in geometry.

Find the ratio of the volumes:

The volumes of the two similar solids are given as 729 cubic inches and 125 cubic inches. To find the ratio of the volumes, we divide the volume of the larger solid by the volume of the smaller solid:

Volume Ratio = Volume of Larger Solid / Volume of Smaller Solid

Volume Ratio = 729 inches $^3$ / 125 inches $^3$

Volume Ratio = 729/125

This ratio tells us how many times greater the volume of the larger solid is compared to the smaller solid. It is a crucial step in establishing the relationship between the two solids and will be used to determine the scale factor.
Determine the scale factor (k):

Since the ratio of the volumes is k $^3$ , we need to find the cube root of the volume ratio to determine the scale factor k.

k $^3$ = 729/125

k = ∛(729/125)

k = ∛729 / ∛125

k = 9/5

The scale factor k = 9/5 represents the ratio of the linear dimensions of the larger solid to the smaller solid. This means that any linear measurement (such as side length, radius, or height) of the larger solid is 9/5 times the corresponding measurement of the smaller solid. The scale factor is a critical link between the two solids, enabling us to relate their surface areas and other properties.
Find the ratio of the surface areas:

The ratio of the surface areas is the square of the scale factor, k $^2$ .

Surface Area Ratio = k $^2$

Surface Area Ratio = (9/5) $^2$

Surface Area Ratio = 81/25

This ratio indicates how many times greater the surface area of the larger solid is compared to the surface area of the smaller solid. It's a direct consequence of the similarity relationship and the scale factor we calculated earlier. Understanding the surface area ratio is essential for finding the unknown surface area of the larger solid.
Calculate the surface area of the larger solid:

We know the surface area of the smaller solid is 74.32 square inches. We can use the surface area ratio to find the surface area of the larger solid.

Surface Area Ratio = Surface Area of Larger Solid / Surface Area of Smaller Solid

81/25 = Surface Area of Larger Solid / 74.32 inches $^2$

Surface Area of Larger Solid = (81/25) * 74.32 inches $^2$

Surface Area of Larger Solid = 2. 24 * 74.32 inches $^2$

Surface Area of Larger Solid = 241.3072 inches $^2$

To round to the nearest hundredth:

Surface Area of Larger Solid ≈ 241.31 inches $^2$

This final calculation gives us the surface area of the larger solid, which is approximately 241.31 square inches. By using the principles of similarity and the relationships between volumes, scale factors, and surface areas, we have successfully solved the problem. This methodical approach demonstrates the power of geometric concepts in practical applications.

In conclusion, by understanding the relationships between similar solids, specifically the ratios of their volumes and surface areas, we were able to determine the surface area of the larger solid. The key steps involved finding the ratio of the volumes, calculating the scale factor, determining the ratio of the surface areas, and finally, computing the surface area of the larger solid. This problem highlights the importance of grasping the fundamental concepts of similarity in geometry. Mastering these principles allows us to solve a wide range of problems involving three-dimensional figures. The ability to apply these concepts is not only crucial for academic success but also for various real-world applications in fields such as architecture, engineering, and design. Through careful analysis and step-by-step calculations, we can confidently tackle geometric challenges and arrive at accurate solutions. The principles discussed in this article provide a solid foundation for further exploration in the fascinating world of geometry.

The surface area of the larger solid is approximately 241.31 square inches.