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Volume of Prisms and Pyramids
Students calculate and compare volumes of prisms and pyramids using base area and perpendicular height.
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Beginner
15–20 minutes
Volume & Area
Volume & Area
Page 6 of 8
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Volume of Prisms and Pyramids
At a Glance
Binder SectionVolume & Area
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
How do one-third relationships connect pyramids and cones to prisms and cylinders?
Build Understanding
Volume measures three-dimensional space in cubic units. Every prism uses V = Bh, where B is the area of the base and h is the perpendicular height.
A pyramid with the same base and height has one-third the prism volume: V = one-third Bh. First calculate B using the correct two-dimensional formula.
Visual Model
4Thousands
3Hundreds
2Tens
7Ones
4,327 = 4,000 + 300 + 20 + 7
Goals
Learning Targets
- Find base area B first.
- Use perpendicular height.
- Prism: V = Bh.
- Pyramid: V = 1/3 Bh.
- Use cubic units.
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Interactive Vocabulary
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Reveal one step at a time
Worked Examples
Example 1
Cone r=4,h=9
1
V=1/3π(4²)(9)=48π cubic units.
Example 2
Sphere r=3
1
V=4/3π(27)=36π cubic units.
Ask and explain
Common Questions
Is B a side length? No, B is base area.
Why one-third for pyramids? Three matching pyramids can fill a related prism.
Can the base be triangular? Yes.
Notice and correct
Common Misconceptions
Omitting the one-third factor for cones or pyramids
Compare with a matching cylinder or prism.
Using slant height in a volume formula
Volume uses perpendicular height.
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Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
Find a pyramid’s volume when B=30 and h=12.
2
We Do
Guided Practice
Rectangular prism 4x6x9: V=216 cubic units.
Triangular prism B=15,h=8: V=120.
Square pyramid base side 10,height 12: V=400.
Prism B=22,height 5.5: V=121.
Pyramid B=54,height 7: V=126.
3
You Do
Independent Practice
Find a pyramid’s volume when B=30 and h=12.
Create and solve a second example with different values.
4
Stretch Your Thinking
Challenge & Real-World Practice
Compare a cone, cylinder, and sphere that share appropriate dimensions. Determine and explain their exact volume ratios.
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