Building a Volume Formula for a Pyramid

ID: rohoj-jutoh
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSG-GMD.A.1HSG-GMD.B.4
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7 questions

Building a Volume Formula for a Pyramid

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Cover Your Bases

For each solid, draw and label a prism or cylinder that has a base congruent to the solid’s and a height equal to the solid’s.

1)

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2)

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Comparing Cross Sections

Each solid in the image has height 6 units. The area of each solid’s base is 10 square units. A cross section has been created in each by dilating the base using the apex as a center with scale factor k=0.5k = 0.5.

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3) Calculate the area of each of the 3 cross sections.

4) Suppose a new cross section was created in each solid, all at the same height, using some scale factor kk. What would the area of each cross section be?

5) Explain your reasoning.

6) What does this information about cross sections tell you about the volumes of the 3 solids?

7) Calculate the volume of each of the solids.