Similar Solids Scale Factors, Surface Area Ratios, and Volume Ratios Study Guide

Conceptual Overview of Similarity: Two solids are considered similar if they have the same shape and their corresponding linear dimensions are proportional.
Relationship Between Ratios: In similar solids, different parameters relate to each other through the scale factor, denoted as $k$ . If the ratio of corresponding lengths (Scale Factor) is $a:b$ , the following relationships apply: * Scale Factor ( $k$ ): $a:b$ * Surface Area Ratio ( $k^2$ ): $a^2:b^2$ * Volume Ratio ( $k^3$ ): $a^3:b^3$

Determining Dimensional Ratios: Based on a comparison between a larger prism and a smaller prism with the following dimensions: * Large Prism Dimensions: Linear measurements of $18\,\text{in}$ and $9\,\text{in}$ . * Small Prism Dimensions: Linear measurements of $4\,\text{in}$ and $2\,\text{in}$ .
Scale Factor Calculation: * The ratio is established by comparing corresponding sides: $18:4$ or $9:2$ . * The simplified scale factor is $9:2$ .
Surface Area Ratio Calculation: * Formula: $k^2 = 9^2:2^2$ * Result: $81:4$
Volume Ratio Calculation: * Formula: $k^3 = 9^3:2^3$ * Result: $729:8$

Case Study - Solid A and Solid B: * Volume of Solid A: $28\,\text{m}^3$ * Volume of Solid B: $1,792\,\text{m}^3$
Step 1: Simplify the Volume Ratio ( $k^3$ ): * $\frac{28\,\text{m}^3}{1,792\,\text{m}^3}$ * Divide by $7$ : $\frac{4\,\text{m}^3}{256\,\text{m}^3}$ * Divide by $4$ : $\frac{1\,\text{m}^3}{64\,\text{m}^3}$ * Volume Ratio = $1\,\text{m}^3:64\,\text{m}^3$
Step 2: Determine the Scale Factor ( $k$ ): * Calculate the cube root of the volume ratio units: $\sqrt[3]{1\,\text{m}^3}:\sqrt[3]{64\,\text{m}^3}$ * Scale Factor = $1\,\text{m}:4\,\text{m}$
Step 3: Calculate the Surface Area Ratio ( $k^2$ ): * Formula: $1^2:4^2$ * Result: $1\,\text{m}^2:16\,\text{m}^2$

Problem Statement: Two similar prisms are provided. The smaller prism has a surface area of $80\,\text{mm}^2$ , and the larger prism has a surface area of $245\,\text{mm}^2$ . Given that the volume of the larger prism is $1,029\,\text{mm}^3$ , calculate the volume of the smaller prism.
Step 1: Simplify the Surface Area Ratio ( $k^2$ ): * $\frac{80\,\text{mm}^2}{245\,\text{mm}^2}$ * Divide by $5$ : $\frac{16\,\text{mm}^2}{49\,\text{mm}^2}$
Step 2: Determine the Scale Factor ( $k$ ): * $\sqrt{16\,\text{mm}^2}:\sqrt{49\,\text{mm}^2}$ * Scale Factor = $4\,\text{mm}:7\,\text{mm}$
Step 3: Calculate the Volume Ratio ( $k^3$ ): * Formula: $(4\,\text{mm})^3:(7\,\text{mm})^3$ * Volume Ratio = $64\,\text{mm}^3:343\,\text{mm}^3$
Step 4: Solve for Unknown Volume ( $x$ ): * Set up the proportion: $\frac{64\,\text{mm}^3}{343\,\text{mm}^3} = \frac{x}{1,029\,\text{mm}^3}$ * Cross-multiply: $343x = 64 \times 1,029$ * Solve for $x$ : $\frac{65,856}{343} = 192$ * Final Result: The volume of the smaller prism is $192\,\text{mm}^3$ .

Definition of Factor: A factor is a positive integer that is being multiplied by some parameter of a formula or existing value (e.g., " $25$ times larger").

General Formula for Volume: $V = \pi r^2 h$
Scenario 1 (Standard Cylinder): * Parameters: Radius ( $r$ ) = $2$ , Height ( $h$ ) = $4$ . * Calculation: $V = \pi(2)^2(4)$ * Simplification: $V = \pi(4)(4)$ * Final Volume: $16\pi$
Scenario 2 (Modified Radius): * Parameters: Radius ( $r$ ) = $10$ , Height ( $h$ ) = $4$ . * Calculation: $V = \pi(10)^2(4)$ * Simplification: $V = \pi(100)(4)$ * Final Volume: $400\pi$
Comparative Observation: When the radius is increased from $2$ to $10$ (a factor of $5$ ), the volume increases by a factor of $5^2 = 25$ . This is confirmed by $16\pi \times 25 = 400\pi$ .