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Math · Geometry · Unit 5

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.

Worksheets

Every sheet has an answer key. Free to print, no login.

Unit
📝 Worksheet ✅ Answer guide
Concept
📝 Worksheet ✅ Answer key
Practice
Set 1
📝 Worksheet ✅ Answer key
Set 2
📝 Worksheet ✅ Answer key
Advanced
Set 1
📝 Worksheet ✅ Answer key
Set 2
📝 Worksheet ✅ Answer key

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
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Adapted from the Illustrative Mathematics curriculum (v.360), licensed CC BY-NC 4.0. © Illustrative Mathematics.