Unit 7 · Core unit
Three-dimensional measurement and a look at how geometry itself is built from axioms. Composite solids are decomposed into parts whose measures add.
What a strong answer looks like
A strong answer decomposes a composite solid explicitly and keeps units consistent throughout.
Know
Each solid has its own formulas, and a composite solid is handled by adding the volumes of its parts.
Apply
Split a composite into named solids, compute each, then add.
Watch out
Adding surface areas of parts without removing the joined faces, which double counts.
Study move
Find the volume of a cylinder topped by a hemisphere.
Know
A plane through a solid produces a cross section whose shape depends on the angle. Scaling a solid changes area by the square and volume by the cube of the factor.
Apply
Apply the square and cube rules rather than scaling every measure linearly.
Watch out
Doubling a dimension and expecting volume to double.
Study move
State how surface area and volume change when a solid is scaled by three.
Know
Euclidean geometry follows from a chosen set of postulates. Changing one produces a consistent but different geometry.
Apply
Identify which results depend on the parallel postulate.
Watch out
Assuming Euclidean results hold in every geometry.
Study move
Name one result that fails when the parallel postulate is changed.
Modeling drill
A cylinder of radius 3 and height 8 is topped by a hemisphere of the same radius. Find the total volume, showing each part.
