Side Length, Volume, and Surface Area of Similar Solids

• ## 6.3.2 Side Length, Volume, and Surface Area of Similar Solids

The user can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships among edge lengths, surface areas, and volumes.

### Instructions

To change the size of prism A, adjust the blue slider or drag the upper right-hand vertex (red circle) in a diagonal motion. Click on the Show/Hide Surface Area button to show or hide the graph, values, and ratio of the surface areas. Click on the Show/Hide Volume button to show or hide the graph, values, and ratio of the volumes.

### Exploration

Your task is to investigate how changing the lengths of the sides of a rectangular prism affects the volume and surface area of the prism. First notice that the two given rectangular prisms are congruent (equal angles and equal sides). Now change the size of the purple prism (A) by grabbing the red dot and dragging it diagonally. Are the two prisms still congruent? Are they similar? Click on "Show Volume." Change the size of prism A again and observe the changes in the measurements. What is being depicted in the graph? What can you say about the relationship between the side lengths and the volume of a rectangular prism?
Next, click on "Show Surface Area" and "Hide Volume." Again, change the size of prism A and observe the changes in measurement. What is being depicted in the graph? What can you say about the relationship between the side lengths and the surface area of a rectangular prism?

### Discussion

As students experiment with different ratios of side lengths (different scale factors), they have the opportunity to observe and interpret the changes in the volume and surface-area data. Students should be encouraged to compare the scale factor to the ratio of the volumes and to the ratio of the surface areas and Side Length and Area of Similar Figures to look for patterns. Teachers can help students consider the relationships between scale factor, side length, volume, and surface area by asking questions like, What is being depicted in the "Volume" graph? Similar questions can be asked about the "Surface Area" graph. Creating tables of values for scale factor, side length, surface area, and volume may help students organize their information and more easily examine how a change in side length affects surface area and volume.
Students may notice a difference in the appearance of the graphs. It is important to focus on why the relationship between side length and volume is cubic whereas the relationship between side length and surface area is quadratic. It contributes to students' understanding of the measures of length, surface area, and volume, and it can help students learn about scale factors. Teachers can help students take notice of that difference by asking questions like, Why does the graph depicting the relationship between side length and surface area differ from the graph depicting the relationship between side length and volume? Teachers can then help students understand that difference by asking questions like, Compare the volume scale factor and the surface-area factor. What is the relationship between those factors? How are those factors represented in the "Volume" and "Surface Area" graphs?

### Take Time to Reflect

The Geometry Standard states that "in grades 6–8 all students should understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects."

• How would these activities help students develop an understanding of surface area and volume of similar solids?
• How does the dynamic nature of the figure help support the development of this type of understanding?
• What other concepts related to surface area and volume are important for students in grades 6–8 to understand?

### Objectives and Standards

NCTM Standards and Expectations
• Geometry / Measurement
• Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• 6-8
• Geometry
• Measurement