Reasoning · Grade 6-2 Solids of Revolution and Their Cross-Sections
Problem
Distance from the axis becomes the radius.
This is the definition of a circle again -- all points a fixed distance from a centre. Spinning turns 'distance from the axis' into 'radius', which is why the solid is round when you look down on it.
7.G.B.4Visualize Spatial RelationshipsEvery point of the figure sweeps out a circle whose radius is its distance from the axis.
Spinning keeps a point the same distance from the axis the whole way round, which is exactly what a circle is.
Each height of the figure gives one width, so the solid is fully described by reading the widths height by height.
Cut the trapezoid into a rectangle and a right triangle.
A right trapezoid is a rectangle with a right triangle stuck on, and cutting at the height where the short side ends is what separates them.
4.G.A.2Identify SubproblemsThe rectangle sweeps a cylinder.
A rectangle spun about one of its sides is the standard picture of a cylinder -- like a rectangle of card taped to a pencil and twirled.
7.G.A.3Visualize Spatial RelationshipsThe triangle sweeps a cone.
A right triangle spun about its vertical leg is the standard picture of a cone; the slanting side becomes the sloping surface and the top corner becomes the point.
7.G.A.3Visualize Spatial RelationshipsTogether they give a cone on a cylinder.
Drawing hidden edges dashed is the usual convention for solids, and it is what shows the reader that the join between cone and cylinder is a full circle.
4.G.A.1Draw A DiagramFor (2), slice through the axis.
Slicing a solid of revolution down the axis undoes the spinning: the axis is a line of symmetry of the cut, so taking half of the cut recovers the shape you started from.
7.G.A.3Work BackwardsRead off the width at each height.
The hole is what pushes the figure away from the axis: the empty tube in the middle of the solid is exactly the empty gap between the figure and the axis.
7.G.A.3Identify SubproblemsHalf of that face is the original shape.
The axis behaves like a mirror line for the sliced solid, so drawing just one side of it is enough -- the spin will supply the other side and everything in between.
4.G.A.3Draw A DiagramSpinning it back gives the same solid.
Matching each edge of the flat figure to one surface of the solid is the reliable check, because a missing or extra edge shows up immediately as a missing or extra face.
7.G.A.3Visualize Spatial RelationshipsSpinning turns 'how far from the line' into 'how wide the solid is' -- so a piece touching the line makes solid middle, and a piece held back makes a hole.