Side Length and Area of Similar Figures

• ## 6.3.1 Side Length and Area of Similar Figures

The user can manipulate the side lengths of one of two similar rectangles and the scale factor to learn about how the side lengths, perimeters, and areas of the two rectangles are related.

### Instructions

To change the dimensions of the rectangle A (green), click and drag either the top, left vertex (red circle) or the lower, right vertex. To change the size of rectangle B (blue), click and drag the red slider to adjust the scale factor.

Click on the Graphs button to display a graphical representation of the ratio of areas and ratio of perimeters. Click on the Measures button to show the numerical values for perimeter, area, and their associated ratios.

### Exploration

How are the perimeters, areas, and side lengths of similar rectangles related? To change the size and shape of the green rectangle, grab and drag one of the red vertices (corners). Notice that the blue and green rectangles are and remain similar (congruent angles, proportional sides). Change the size of the blue rectangle by adjusting the scale factor. Observe the changes in the measurements.
Now click on Graphs. Again adjust the shape of the green rectangle and the size of the blue rectangle and observe the changes in the measurements and the graphs. What is being shown in the graphs?

### Discussion

As students experiment with rectangles of different shapes and different scale factors relative to side lengths, they have the opportunity to observe and interpret the changes in the perimeter and area. Students should be encouraged to look at what happens for scale factors that are whole numbers. Teachers can help students consider the relationships between perimeters of similar figures by prompting them with questions like, What mathematical relationship is being depicted in the "Ratio of Perimeters" graph? A similar question can be asked about the "Ratio of Areas" graph.
Students may notice a difference in the appearance of the graphs. Focus on why the relationship between the scale factor and the ratio of perimeters of similar rectangles is linear, whereas the relationship between the scale factor and the ratio of areas of similar rectangles is nonlinear. These explorations contribute to students' understanding of the relationship between side lengths, perimeter, and area in conjunction with scale factors. Teachers can help students see the differences in these relationships by asking questions like, Why does the graph depicting the relationship between scale factor and perimeter have a different shape from the graph depicting the relationship between scale factor and area? Teachers can help students develop their understanding by considering questions such as, Compare the ratio of the perimeters with the ratio of the areas for rectangles with various scale factors. What is the relationship between those two ratios?
It may also be beneficial for students to organize their data in a table. For various scale factors, record side lengths, perimeter, and area. This format may help students organize their information and assist in their developing an understanding of the relationship between scaling and perimeter and area.

### 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 a better understanding of perimeter and area with respect to scaling?
• How does the dynamic nature of the figure help support student's development of this understanding?
• What other related concepts could be developed from this investigation?

### Acknowledgments

Special thanks to Nick Jackiw for his timely work and keen insights in creating this applet and to Key Curriculum Press for allowing the use of JavaSketchpad™.

### 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