Sphere Slice Teacher Guide Sphere Slice Showdown
Teacher Facilitation Guide
Geometry Tier 2
Objective
Students will informally derive the volume of a sphere using Cavalieri’s Principle by comparing its cross-sectional areas to a cylinder with an inverted cone removed at various heights.
Standard Alignment
Colorado HS.G-GMD.A.2: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
Required Materials
Two stacks of coins/poker chips
Modeling clay (Play-Doh)
Fishing line (for slicing)
Clear acrylic sphere/cylinder set
Intervention Key
Focus on the Area Equality . If the areas are the same at every height, the total volume must be the same.
Watch for students confusing radius and height during the algebraic step.
Instructional Routine
1
Hook: The Coin Stack (5 mins)
Display a straight stack of 20 quarters and a slanted stack of 20 quarters. Ask: "Do they have the same volume?" Connect this to Cavalieri's Principle : Same cross-section area + Same height = Same Volume.
2
The Comparison Solid (10 mins)
Explain that we compare a hemisphere of radius R to a cylinder of radius R and height R with a cone removed. This is the hardest conceptual leap. Use the clay models here to show the "ring" vs "circle" cross-sections.
The Mathematical Proof Walkthrough
Step-by-Step Questioning Strategy
Q1: "In the hemisphere, what shape is the cross-section at height h?"
Answer: A circle. Using Pythagorean theorem, its radius is \(\sqrt{R^2 - h^2}\). Area = \(\pi(R^2 - h^2)\).
Q2: "In the hollowed-out cylinder, what shape is the cross-section at height h?"
Answer: A ring (annulus). Outer area is \(\pi R^2\). Inner area (cone) is \(\pi h^2\). Ring area = \(\pi R^2 - \pi h^2\).
Q3: "Are these areas equal?"
Answer: Yes! Both simplify to \(\pi R^2 - \pi h^2\). Therefore, Volumes must be equal.
Scaffolding for Tier 2 Students
Visual Support
Use colored water to fill a hollow hemisphere and pour it into the "hollowed cylinder" model. Seeing the water levels match reinforces the Cavalieri principle before the math starts.
Algebraic Support
Provide a pre-filled Pythagorean diagram showing \(r^2 + h^2 = R^2\) for the sphere slice. Many students struggle to connect the triangle to the circular cross-section.
Success Criteria
Students can explain why two differently shaped stacks of the same base area and height have the same volume.
Students can identify the circle and the annulus as the cross-sections for the comparison.
Students can set up the volume equation: \(V_{hemisphere} = V_{cylinder} - V_{cone}\).
Sphere Volume Slides Geometry Intervention
Sphere Slice Showdown
Deriving volume through the magic of Cavalieri's Principle.
Cross-Sections
Volume
The Big Idea
Cavalieri's Principle
"If two solids have the same height and the same cross-sectional area at every level, they have the same volume."
Think of two stacks of 20 quarters. One is perfectly straight, the other is leaning. Do they contain the same amount of metal?
Same height. Same layers. Same Volume.
The Comparison
Solid A: Hemisphere
A sphere with radius \(R\) cut in half.
Solid B: "The Ring Cylinder"
A cylinder (radius \(R\), height \(R\)) with a cone removed.
To use Cavalieri's Principle, we need to show their areas match at every height (h).
Layer 1: The Sphere
R h r
Finding the Radius of the slice (\(r\)):
Using Pythagoras: \(r^2 + h^2 = R^2\)
Therefore, \(r^2 = R^2 - h^2\)
Cross-Section Area
Area = \(\pi r^2\)
Area = \(\pi(R^2 - h^2)\)
Layer 2: The Ring
R
h
Top View of the cylinder slice at height h
The Annulus (Ring) Area:
Subtract the inner circle from the outer circle.
Area = \(\pi R^2 - \pi h^2\)
Cross-Section Area
Area = \(\pi(R^2 - h^2)\)
IT'S A MATCH!
The Conclusion
Since the areas match, the volumes must match:
Hemisphere \(V_{hemi}\)
=
Cylinder - Cone \(\pi R^3 - \frac{1}{3}\pi R^3\)
Therefore, for a full sphere:
Volume = \(\frac{4}{3} \pi r^3\)
Sphere Volume Worksheet Sphere Volume Blueprint
Guided Exploration & Practice
Name:
Date:
Part 1: The Observation
Recall Cavalieri's Principle : If two solids have the same and the same cross-sectional at every height, they have the same volume .
Reflect:
Why do these two stacks have the same volume, even though they look different?
Part 2: The Math Blueprint
The Hemisphere Slice
R h r
Step 1: Find slice radius \(r\) in terms of \(R\) and \(h\).
\(r^2 = \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\)
Slice Area Calculation:
\(Area = \pi (\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_})\)
The "Ring" Slice
Step 2: Area of the Ring (Annulus).
Outer Circle Radius = \(R\). Inner Circle Radius = \(h\).
Slice Area Calculation:
\(Area = \pi (\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}) - \pi (\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_})\)
The Matchup
Look at both Slice Area equations above. Are they mathematically identical?
YES
NO
Part 3: Construction Crew Practice
1. A sphere has a radius of 6 cm. Using the formula \(V = \frac{4}{3} \pi r^3\), calculate its exact volume in terms of \(\pi\).
2. If you cut the sphere from Problem 1 in half, what is the volume of the resulting hemisphere?
Challenge Case
A cylinder with a height of 10 and a radius of 10 has a cone removed from its center (the cone also has height 10 and radius 10). Based on Cavalieri's Principle, what is the volume of this resulting hollowed solid? (Hint: Compare it to a hemisphere).
Sphere Slice Exit Ticket Quick Check: Slice & Dice
Exit Ticket
Sphere Volume & Cavalieri's Principle
Student Name
1 State Cavalieri's Principle in your own words. What two conditions must be met for two solids to have the same volume?
2 A hemisphere of radius 5 cm is being compared to a "ringed cylinder" of radius 5 cm and height 5 cm. At a height of \(h = 3\) cm, what is the area of the cross-section for both solids?
Show your work below. Use \(\pi\) in your final answer.
Hemisphere Slice
Ring Slice
3 Why do we subtract a cone from the cylinder to prove the sphere's volume, rather than just using a plain cylinder?
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