Volumes of Similar Figures Game/Worksheet

Volumes of Similar Figures Game/Worksheet
Welcome to the Volumes of Similar Figures Challenge! This game helps the student to explore how expanding the length, width, and height of an object affects its volume. Work with cubes, cylinders, cones, and spheres to master the cubed scale factor (k³) rule. Scroll down the page for a more detailed explanation.

 

 

How to Play

1. Analyze the Figures: On the screen, you will see two mathematically similar 3D shapes side-by-side: Figure 1 and Figure 2.
   

2. Identify the Given Clues: Look at the numbers provided on the shapes. The game will show you:
   The linear measurement (like an edge, radius, or height) for both figures.
   The total volume for only one of the figures.
   

3. Calculate the Missing Volume: Find the missing total volume marked with a question mark (?).
   

4. Submit and Verify: Type your answer into the box and click “Verify Space Volume.” A correct answer awards you 100 points and moves you to the next level. If you get stuck, a clear step-by-step math breakdown will guide you through the solution!
   

How the Math Works
The secret to mastering this game is a simple rule of geometry: when 3D figures are similar, their volumes scale by the cube of their linear dimensions.

Step 1: Find the Scale Factor (k)
First, compare the two matching lines by dividing the dimension of Figure 2 by the dimension of Figure 1. This gives you your basic linear scale factor (k):

\(k = \frac{\text{Dimension}_2}{\text{Dimension}_1}\)

Step 2: Cube the Scale Factor (k³)
Because volume is three-dimensional (Length × Width × Height), you must multiply the scale factor by itself three times to see how the volume changes:

\(\text{Volume Scale Factor} = k^3\)

For example: If a toy’s edges are multiplied by 2, its volume doesn’t just double—it grows by 2 × 2 × 2, meaning it can hold 8 times more inside.

Step 3: Solve for the Unknown Volume
Set up your proportional equation to find your missing number:

\(\frac{\text{Volume}_2}{\text{Volume}_1} = \left(\frac{\text{Dimension}_2}{\text{Dimension}_1}\right)^3\)

If you are looking for the larger volume (Figure 2), multiply the smaller volume by k³.
If you are looking for the smaller volume (Figure 1), divide the larger volume by k³.

Volumes of Similar Figures

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