Scaling Principle for Volumes

New York State Common Core Math Geometry, Module 3, Lesson 9

Worksheets for Geometry

Student Outcomes

• Students understand that given similar solids A and B so that the ratio of their lengths is a:b, then the ratio of their volumes is a³:b³.
• Students understand that if a solid with volume V is scaled by factors of r, s, and t in three perpendicular directions, then the volume is multiplied by a factor of r·s·t so that the volume of the scaled solid is (rst)V.

Scaling Principle for Volumes

Classwork

Opening Exercise

a. For each pair of similar figures, write the ratio of side lengths 𝑎: 𝑏 or 𝑐: 𝑑 that compares one pair of corresponding sides. Then, complete the third column by writing the ratio that compares the areas of the similar figures. Simplify ratios when possible.
b. i. State the relationship between the ratio of sides 𝑎: 𝑏 and the ratio of the areas Area(𝐴): Area(𝐵).
ii. Make a conjecture as to how the ratio of sides 𝑎: 𝑏 will be related to the ratio of volumes Volume(𝑆): Volume(𝑇). Explain.
c. What does it mean for two solids in three-dimensional space to be similar?

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Exercises

1. Each pair of solids shown below is similar. Write the ratio of side lengths 𝑎: 𝑏 comparing one pair of corresponding sides. Then, complete the third column by writing the ratio that compares volumes of the similar figures. Simplify ratios when possible.
2. Use the triangular prism shown below to answer the questions that follow.
   a. Calculate the volume of the triangular prism.
   b. If one side of the triangular base is scaled by a factor of 2, the other side of the triangular base is scaled by a factor of 4, and the height of the prism is scaled by a factor of 3, what are the dimensions of the scaled triangular prism?
   c. Calculate the volume of the scaled triangular prism.
   d. Make a conjecture about the relationship between the volume of the original triangular prism and the scaled triangular prism.
   e. Do the volumes of the figures have the same relationship as was shown in the figures in Exercise 1? Explain.
3. Use the rectangular prism shown to answer the questions that follow.
   a. Calculate the volume of the rectangular prism
   b. If one side of the rectangular base is scaled by a factor of 1/2, the other side of the rectangular base is scaled by a factor of 24, and the height of the prism is scaled by a factor of 1/3, what are the dimensions of the scaled rectangular prism?
   c. Calculate the volume of the scaled rectangular prism.
   d. Make a conjecture about the relationship between the volume of the original rectangular prism and the scaled rectangular prism.
4. A manufacturing company needs boxes to ship their newest widget, which measures 2 × 4 × 5 in³. Standard size boxes, 5-inch cubes, are inexpensive but require foam packaging so the widget is not damaged in transit. Foam packaging costs $0.03 per cubic inch. Specially designed boxes are more expensive but do not require foam packing. If the standard size box costs $0.80 each, and the specially designed box costs $3.00 each, which kind of box should the company choose? Explain your answer.

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