CCSS.Math.Content.HSG-GMD.A.2
The Standard
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students compare two solids by slicing them at the same height and finding each cross-sectional area. They use matching slice areas to justify equal volumes and derive volume formulas.
What Mastery Looks Like
- A student can sketch a sphere and a cylinder with two cones removed, then compare slices at the same height. The student uses those equal areas to derive the sphere volume formula.
Common Misconceptions
- Students may compare only one slice instead of slices at every height. They may confuse cross-sectional area with surface area or assume solids with equal heights have equal volumes.
How to Assess It
- Exit ticket: A sphere of radius r is compared with a radius-r, height-2r cylinder after two height-r cones are removed. At distance x from the center, show the slice areas match, then find the sphere’s volume.
Lesson moves
Ways to Teach It
Use compasses to cut a sphere disk and comparison annulus for three heights, then verify their areas match.
Write an explanation answering: Why do equal cross-sectional areas at every shared height mean two solids have equal volume?
Match cards showing solids, slice-area expressions, and volume formulas, then justify each matched set before scoring a point.
Model a spherical water tank as an equal-volume cylinder with two cones removed, then calculate its capacity from a given radius.
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Printable HSG-GMD.A.2 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-GMD.A.2, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSG-GMD.A.1
Arguments for cylinder, pyramid, and cone volumes support Cavalieri comparisons and the sphere formula through cross-sections and known solid volumes.
- CCSS.Math.Content.HSG-GMD.B.4
Cavalieri arguments compare matching cross-sections, so recognizing slices of solids supports deriving sphere and other volume formulas.
- CCSS.Math.Content.7.G.B.4
Cavalieri arguments for sphere volume require computing circular cross sections with the circle area formula.
Keep exploring
Related Standards
- CCSS.Math.Content.HSG-GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- CCSS.Math.Content.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- CCSS.Math.Content.HSG-GMD.A
Explain volume formulas and use them to solve problems
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