Finding Surface Area Of Similar Solids A Step By Step Guide

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 their corresponding angles are equal, and their corresponding sides are proportional. This proportionality extends to their surface areas and volumes, creating a fascinating interplay between these measurements. In this article, we will delve into the relationship between the volumes and surface areas of similar solids, providing a step-by-step guide on how to solve problems involving these concepts. We will focus on a specific problem: The volumes of two similar solids are 210 m³ and 1,680 m³. The surface area of the larger solid is 856 m². What is the surface area of the smaller solid? This problem will serve as a practical example to illustrate the principles and techniques involved. We will explore the underlying mathematical concepts, discuss the formulas used, and demonstrate the problem-solving process in a clear and concise manner. By the end of this guide, you will have a solid understanding of similar solids and be equipped to tackle similar problems with confidence. This knowledge is not only valuable for academic pursuits but also has practical applications in various fields, such as architecture, engineering, and design, where scaling and proportions are essential considerations. So, let's embark on this geometrical journey and unravel the secrets of similar solids.

Understanding Similar Solids

Before we tackle the problem at hand, let's lay a solid foundation by understanding the properties of similar solids. Similar solids, as mentioned earlier, are three-dimensional shapes that maintain the same form but differ in size. Imagine two cubes, one smaller and one larger; if their corresponding sides are in proportion, they are similar. This proportionality is the key to unlocking the relationship between their surface areas and volumes. The ratio of their corresponding linear dimensions, such as side lengths, radii, or heights, is called the scale factor. This scale factor is the cornerstone upon which we build our understanding of how surface areas and volumes relate. For example, if one cube has sides twice as long as another, the scale factor is 2. This simple ratio has profound implications for the surface areas and volumes of the cubes. The surface area, being a two-dimensional measure, scales with the square of the scale factor. This means that if the scale factor is 2, the larger cube's surface area is 2² = 4 times that of the smaller cube. This is because surface area is calculated using the product of two dimensions, and each dimension is scaled by the scale factor. The volume, a three-dimensional measure, scales with the cube of the scale factor. Continuing with our example, the larger cube's volume is 2³ = 8 times that of the smaller cube. This is because volume is calculated using the product of three dimensions, and each dimension is scaled by the scale factor. These relationships are fundamental to solving problems involving similar solids. Knowing the scale factor allows us to directly calculate the ratio of surface areas and volumes, and vice versa. Understanding these concepts is not just about memorizing formulas; it's about grasping the underlying principles of how scaling affects different dimensions. This deeper understanding will enable you to tackle a wider range of problems and apply these concepts in various contexts.

Key Ratios: Scale Factor, Surface Area Ratio, and Volume Ratio

To effectively solve problems involving similar solids, it's crucial to understand and apply the key ratios that govern their relationships. These ratios provide a direct link between the dimensions, surface areas, and volumes of similar figures. The first and most fundamental ratio is the scale factor. As we discussed earlier, the scale factor is the ratio of corresponding linear dimensions of the similar solids. This could be the ratio of their side lengths, radii, heights, or any other corresponding linear measurement. If we denote the scale factor as k, then the ratio of any corresponding linear dimension of the larger solid to the smaller solid is k. This scale factor acts as the foundation for determining the other ratios. The second key ratio is the surface area ratio. The surface area ratio is the ratio of the surface areas of the two similar solids. It's directly related to the scale factor. The surface area ratio is equal to the square of the scale factor, k². This means that if you know the scale factor, you can easily find the ratio of their surface areas. Conversely, if you know the ratio of their surface areas, you can find the scale factor by taking the square root of the surface area ratio. This relationship arises because surface area is a two-dimensional measure, and scaling a two-dimensional quantity involves squaring the scale factor. The third crucial ratio is the volume ratio. The volume ratio is the ratio of the volumes of the two similar solids. It's related to the scale factor in a similar way to the surface area ratio, but instead of squaring, we cube the scale factor. The volume ratio is equal to the cube of the scale factor, k³. This means that if you know the scale factor, you can find the ratio of their volumes by cubing it. Similarly, if you know the ratio of their volumes, you can find the scale factor by taking the cube root of the volume ratio. This relationship stems from the fact that volume is a three-dimensional measure, and scaling a three-dimensional quantity involves cubing the scale factor. Mastering these three ratios – the scale factor, the surface area ratio, and the volume ratio – is essential for solving problems involving similar solids. They provide a direct and efficient way to relate the dimensions, surface areas, and volumes of these figures. Understanding how these ratios are derived from the scaling principles further solidifies your grasp of the concepts and enables you to apply them confidently in various problem-solving scenarios.

Problem Breakdown: Volumes and Surface Area

Now that we have a firm grasp of the underlying principles and key ratios, let's dissect the problem at hand. The volumes of two similar solids are 210 m³ and 1,680 m³. The surface area of the larger solid is 856 m². What is the surface area of the smaller solid? This problem presents us with the volumes of two similar solids and the surface area of the larger solid. Our goal is to find the surface area of the smaller solid. To solve this, we'll follow a structured approach, leveraging the relationships between scale factor, surface area ratio, and volume ratio. The first step is to identify the given information. We know the volume of the smaller solid (210 m³), the volume of the larger solid (1,680 m³), and the surface area of the larger solid (856 m²). We need to find the surface area of the smaller solid. The second step is to determine the volume ratio. The volume ratio is the ratio of the volume of the larger solid to the volume of the smaller solid. In this case, the volume ratio is 1,680 m³ / 210 m³ = 8. This ratio tells us that the larger solid is eight times the volume of the smaller solid. The third step is to calculate the scale factor. We know that the volume ratio is the cube of the scale factor (k³). Therefore, to find the scale factor (k), we need to take the cube root of the volume ratio. The cube root of 8 is 2, so the scale factor (k) is 2. This means that the linear dimensions of the larger solid are twice the linear dimensions of the smaller solid. The fourth step is to determine the surface area ratio. The surface area ratio is the square of the scale factor (k²). Since the scale factor is 2, the surface area ratio is 2² = 4. This ratio tells us that the surface area of the larger solid is four times the surface area of the smaller solid. The final step is to calculate the surface area of the smaller solid. We know the surface area of the larger solid (856 m²) and the surface area ratio (4). To find the surface area of the smaller solid, we divide the surface area of the larger solid by the surface area ratio: 856 m² / 4 = 214 m². Therefore, the surface area of the smaller solid is 214 m². By breaking down the problem into these steps, we've systematically utilized the relationships between volumes, surface areas, and the scale factor to arrive at the solution. This structured approach is key to tackling similar problems effectively.

Step-by-Step Solution: Finding the Surface Area

Having broken down the problem and outlined the steps involved, let's now walk through the solution in detail, solidifying our understanding of the process. We'll revisit each step, providing the calculations and reasoning behind them. This step-by-step approach will not only lead us to the correct answer but also reinforce the concepts and techniques we've discussed. Step 1: Identify the given information. We are given the following information:

• Volume of the smaller solid: 210 m³
• Volume of the larger solid: 1,680 m³
• Surface area of the larger solid: 856 m²

Our objective is to find the surface area of the smaller solid. Step 2: Determine the volume ratio. The volume ratio is the ratio of the volume of the larger solid to the volume of the smaller solid. We calculate this as follows:

Volume Ratio = (Volume of Larger Solid) / (Volume of Smaller Solid) = 1,680 m³ / 210 m³ = 8

This tells us that the larger solid is eight times the volume of the smaller solid. Step 3: Calculate the scale factor. The scale factor (k) is the cube root of the volume ratio. We calculate this as follows:

k = ∛(Volume Ratio) = ∛8 = 2

The scale factor is 2, which means that the linear dimensions of the larger solid are twice the linear dimensions of the smaller solid. Step 4: Determine the surface area ratio. The surface area ratio is the square of the scale factor. We calculate this as follows:

Surface Area Ratio = k² = 2² = 4

This indicates that the surface area of the larger solid is four times the surface area of the smaller solid. Step 5: Calculate the surface area of the smaller solid. We can now calculate the surface area of the smaller solid by dividing the surface area of the larger solid by the surface area ratio:

Surface Area of Smaller Solid = (Surface Area of Larger Solid) / (Surface Area Ratio) = 856 m² / 4 = 214 m²

Therefore, the surface area of the smaller solid is 214 m². By meticulously following these steps, we've arrived at the solution. Each step builds upon the previous one, utilizing the relationships between volumes, surface areas, and the scale factor. This systematic approach is a powerful tool for solving problems involving similar solids.

Answer and Options Analysis

Now that we have meticulously worked through the problem and arrived at the solution, let's compare our answer with the given options and discuss why the other options are incorrect. This analysis will not only confirm our solution but also deepen our understanding of the problem-solving process and the potential pitfalls to avoid. Our calculated surface area of the smaller solid is 214 m². Let's examine the given options:

A) 107 m² B) 214 m² C) 1034 m² D) 6848 m²

Comparing our answer with the options, we see that option B) 214 m² matches our calculated value. Therefore, option B is the correct answer. Now, let's analyze why the other options are incorrect:

• Option A) 107 m²: This option is incorrect because it represents half of the correct answer. This might be a result of mistakenly dividing the surface area of the larger solid by the scale factor (2) instead of the surface area ratio (4). This highlights the importance of understanding the distinction between the scale factor and the surface area ratio. The scale factor relates linear dimensions, while the surface area ratio relates surface areas, which are two-dimensional measures. Therefore, we need to use the square of the scale factor for surface area calculations.
• Option C) 1034 m²: This option is significantly larger than the correct answer. This could be a result of mistakenly multiplying the surface area of the larger solid by the surface area ratio (4) instead of dividing. This error suggests a misunderstanding of the inverse relationship between the surface areas of the smaller and larger solids. Since the smaller solid has a smaller surface area, we need to divide the surface area of the larger solid by the surface area ratio to find the surface area of the smaller solid.
• Option D) 6848 m²: This option is the largest and most significantly incorrect answer. This could be a result of a combination of errors, such as multiplying the surface area of the larger solid by the cube of the scale factor (2³ = 8) or misinterpreting the relationships between the volumes and surface areas. This option underscores the importance of carefully reviewing each step and ensuring that the correct operations are performed with the correct values. By analyzing the incorrect options, we gain valuable insights into the common mistakes that can occur when solving problems involving similar solids. This understanding will help us avoid these pitfalls and approach similar problems with greater accuracy and confidence.

Conclusion: Mastering Similar Solids

In conclusion, understanding the relationships between similar solids is a fundamental concept in geometry with wide-ranging applications. In this comprehensive guide, we've explored the key principles governing the surface areas and volumes of similar solids, focusing on the crucial role of the scale factor and its impact on these measurements. We tackled a specific problem: The volumes of two similar solids are 210 m³ and 1,680 m³. The surface area of the larger solid is 856 m². What is the surface area of the smaller solid? Through a structured, step-by-step approach, we successfully determined the surface area of the smaller solid to be 214 m². This involved calculating the volume ratio, finding the scale factor, determining the surface area ratio, and finally, using these ratios to calculate the unknown surface area. We emphasized the importance of understanding the underlying concepts rather than simply memorizing formulas. Grasping the relationship between the scale factor and the surface area and volume ratios is crucial for solving a variety of problems involving similar solids. The surface area ratio is the square of the scale factor, while the volume ratio is the cube of the scale factor. These relationships stem from the fact that surface area is a two-dimensional measure, and volume is a three-dimensional measure. We also analyzed the incorrect answer options, highlighting common mistakes that can occur when solving these types of problems. This analysis provided valuable insights into the potential pitfalls to avoid, such as confusing the scale factor with the surface area ratio or misinterpreting the inverse relationship between the surface areas of the smaller and larger solids. By mastering the concepts and techniques discussed in this guide, you'll be well-equipped to tackle a wide range of problems involving similar solids. This knowledge is not only valuable for academic pursuits but also has practical applications in various fields, such as architecture, engineering, and design. So, continue practicing and applying these principles to further solidify your understanding and unlock the fascinating world of geometry.

Practice Problems

To further solidify your understanding of similar solids and their properties, let's explore some additional practice problems. These problems will challenge you to apply the concepts and techniques we've discussed in different scenarios, reinforcing your problem-solving skills. Working through these problems will not only help you master the material but also build your confidence in tackling more complex geometrical challenges. Remember to approach each problem systematically, identifying the given information, determining the relevant ratios, and applying the appropriate formulas. Don't hesitate to revisit the concepts and examples we've covered in this guide if you need a refresher. The key to success is consistent practice and a willingness to learn from your mistakes. Let's dive into the practice problems:

1. Two similar pyramids have heights of 10 cm and 15 cm. The volume of the smaller pyramid is 200 cm³. What is the volume of the larger pyramid?
2. The surface areas of two similar cones are 36π cm² and 81π cm². The volume of the smaller cone is 108π cm³. What is the volume of the larger cone?
3. Two similar rectangular prisms have side lengths in the ratio of 2:3. The total surface area of the smaller prism is 144 cm². What is the total surface area of the larger prism?
4. A sphere has a radius of 5 cm. Another sphere is similar to the first sphere, and its volume is 8 times larger. What is the radius of the larger sphere?
5. The volumes of two similar cylinders are 64π in³ and 216π in³. The surface area of the smaller cylinder is 40π in². What is the surface area of the larger cylinder?

These practice problems cover a variety of shapes and scenarios, providing a comprehensive review of the concepts we've discussed. Take your time to work through each problem carefully, showing your steps and reasoning. Once you've completed the problems, you can check your answers and review your solutions to identify areas where you may need further practice. The solutions to these problems, along with detailed explanations, will be provided in a separate section. Remember, practice makes perfect, and the more you work with these concepts, the more confident and proficient you'll become in solving problems involving similar solids.