Unit 5: Solid Geometry

Students practice spatial visualization in three dimensions. They sketch cross-sections, study how surface area and volume scale under dilation, derive volume formulas for prisms, cylinders, pyramids, and cones using Cavalieri’s Principle, and apply their understandings to solve problems.

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What this unit covers

A Cross-Sections, Scaling, and Area · 5 lessons

Goals

• Describe the relationships between scale factors and areas, using square root graphs and calculations.
• Generate multiple cross-sections of three-dimensional figures.
• Identify the three-dimensional solid created by rotating a two-dimensional figure, using a linear axis.

Lessons

1. Solids of Rotation
2. Slicing Solids
3. Creating Cross-Sections by Dilating
4. Scaling and Area
5. Scaling and Unscaling

B Scaling Solids · 3 lessons

Goals

• Comprehend that when a solid is dilated by a scale factor of k, its surface area is multiplied by k² and its volume is multiplied by k³.
• Describe the relationships between scale factors and volumes using cube root graphs and calculations.

Lessons

1. Scaling Solids
2. The Root of the Problem
3. Speaking of Scaling

C Prism and Cylinder Volumes · 3 lessons

Goals

• Calculate volumes of solids composed of right and oblique prisms and cylinders.
• Explain connections between ways of finding volumes of prisms and cylinders.

Lessons

1. Cylinder Volumes
2. Cross-Sections and Volume
3. Prisms Practice

D Understanding Pyramid Volumes · 4 lessons

Goals

• Calculate volumes and dimensions of prisms, cylinders, cones, and pyramids.
• Use decomposition and Cavalieri’s Principle to informally justify the volume formula for pyramids.

Lessons

1. Prisms and Pyramids
2. Building a Volume Formula for a Pyramid
3. Working with Pyramids
4. Putting All the Solids Together

E Let’s Put It to Work · 3 lessons

Lessons

1. Surface Area and Volume
2. Volume and Density
3. Volume and Graphing