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
What this standard means
Students need to explain volume formulas by comparing cross-sections of solids at the same height. They should use Cavalieri’s principle to argue that if matching slices have equal area all the way up, the solids have equal volume. For spheres, they often compare slices of a hemisphere to slices from a cylinder with a cone removed.
Mastery looks like a clear sketch, labeled height, matching cross-section areas, and a short argument that connects the slices to total volume. Students often get stuck treating the formula as something to memorize, or they compare widths instead of areas of slices.
Ways to teach it
- Use clay or stackable foam sheets to build two solids with equal cross-section areas at matching heights, then compare their volumes.
- Ask students to explain why matching every horizontal slice by area is stronger than matching just the top, bottom, or height.
- Show a diagram of a hemisphere, cylinder, and cone, then have students label one slice area from each and complete the argument.
- Connect to 3D printing by asking why a printer building equal-area layers would use the same amount of material for two different shapes.
Plan a lesson for CCSS.Math.Content.HSG-GMD.A.2
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Related standards
- CCSS.Math.Content.HSG-GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
- 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