Geometry · 35 minutes
Suggested grades 9–11 · Unit 3: Circles and three-dimensional measurement · Lesson 2 of 2
Volume Comparisons for Solid Designs
Compare equal dimensions without confusing radius, diameter and height.
Mr. S Teaches · Original starter lesson · HS-GEO-06 · Grade range is planning guidance.Watch, notice, discuss
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Volume of a cone | Perimeter, area, and volume | Geometry | Khan Academy
By Khan Academy
Revisit the one-third relationship for a cone, then compare the lesson’s solid designs.
Open on YouTube: Volume of a cone | Perimeter, area, and volume | Geometry | Khan Academy (opens a new tab) ↗Volume of a sphere | Perimeter, area, and volume | Geometry | Khan Academy
By Khan Academy
Review the sphere-volume formula and compare the calculated capacity with the lesson’s cylindrical and conical alternatives; the lesson extends the comparison to pyramids.
Open on YouTube: Volume of a sphere | Perimeter, area, and volume | Geometry | Khan Academy (opens a new tab) ↗How can you compare designs with different shapes while keeping units and dimensions consistent?
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Learning objective
Calculate and compare cylinder, cone, pyramid and sphere volumes.
Before you begin
- Area formulas
- Algebraic substitution
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- volume
- Three-dimensional space measured in cubic units.
- base area
- The area of a selected supporting face.
- scale effect
- How a measurement changes under a size multiplier.
Understand the concept
A cylinder has volume Bh; a cone or pyramid has volume Bh/3. A sphere has volume 4πr³/3. Equal heights alone do not imply equal volumes because base areas may differ. When every length doubles, volume multiplies by eight; this follows directly from the three length factors in the formulas.
Teach it step by step
- Compare a radius-3, height-8 cylinder and cone using the same base area 9π.
- Calculate a square pyramid with base side 6 and height 8.
- Calculate a radius-3 sphere and distinguish its diameter from its radius.
- Double all lengths in one design and confirm the factor of eight algebraically.
Make the thinking visible
Worked example or model response
Cylinder:72π; cone:24π; pyramid:(1/3)(36)(8)=96; sphere:36π, all cubic units. A doubled cylinder has radius 6 and height 16, volume 576π=8(72π).
Guided practice
Find a cone’s height when its radius is 2 cm and volume is 20π cm³.
Independent practice
1. Find a diameter-10, height-3 cylinder’s volume. 2. Find a radius-2 sphere’s volume. 3. Explain how tripling all pyramid dimensions affects volume.
Quick assessment
Why is a sphere with radius 3 not described by 4π(6³)/3?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: 20π=(1/3)π(4)h, so h=15 cm.
- Independent 1:75π cubic units. 2:32π/3 cubic units. 3: Volume multiplies by 27.
- Quick check:6 is the diameter; the sphere formula requires the stated radius 3.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Provide a labeled diagram and a formula-substitution-unit template for each solid.
Extension challenge
Construct a cone and cylinder with equal volume but different heights and radii.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSG.GMD.A.3
Common Core · High school · Geometry
Regular high school mathematics standard
Apply solid-volume formulas and compare scale effects.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.