Reasoning · Grade 6-2 Solids of Revolution and Their Cross-Sections
Problem
A solid of revolution looks the same all round its axis.
This is the one fact everything else rests on, and you can feel it with your hands: spin a cardboard shape taped to a pencil and the blur you see is the solid, perfectly round at every height.
7.G.A.3Create A Physical RepresentationFrom above it is a circle or a ring.
Every horizontal slice is a circle around the same centre, so from above they all sit inside the biggest one — nothing can stick out past the widest part.
7.G.A.3Visualize Spatial RelationshipsFront and side views are always identical.
If the solid looks the same from every horizontal direction, then front, side, back and the other side all give the same outline — you only need to draw it once.
4.G.A.3Look For A PatternThe front view and the side view of a solid of revolution are always identical.
Turning the solid a quarter turn about its axis lands it exactly on itself, so the two viewpoints see the very same shape.
At every height the solid is a circle about the axis, so its width is the same whichever side you look from.
The front view mirrors the profile across the axis.
This is the mirror idea from Grade 4: the axis is the line of symmetry, and the flat shape that was spun is one half of the picture.
4.G.A.3Draw A DiagramCylinder and cone give a rectangle and a triangle.
These two are worth doing first because they are the building blocks: every remaining row is rectangles and triangles and trapezoids stacked up.
7.G.A.3Visualize Spatial RelationshipsThe holed truncated cone shows a ring from above.
The trapezoid is exactly the flat shape that was spun, together with its mirror image — you can see the two slanted sides in the picture of the solid.
4.G.A.2Visualize Spatial RelationshipsThe cone-topped cylinder looks like a house from the front.
Because the two pieces have the same diameter, the join makes no step in the outline — the roof sits flush on the walls, so you get one pentagon and not two separate shapes.
7.G.A.3Draw A DiagramThe three-tier solid looks stepped from the front.
Every change of width in the stack becomes a step in the outline, so counting the pieces tells you how many steps to expect before you draw anything.
7.G.A.3Draw A DiagramThe hourglass also shows a ring from above.
The waist is the narrowest height, and it is exactly where the outline pinches in, so you can read the pinch straight off the picture of the solid.
4.G.A.3Visualize Spatial RelationshipsAll eighteen cells obey the three rules.
A rule that holds in all six rows at once is a much stronger check than looking at one row alone, and it catches a lop-sided drawing immediately.
4.G.A.3Look For A PatternAnything made by spinning looks the same from every side, so front and side are one drawing — and from above it is always a circle, with a hole in the middle only if the hole goes right through.