Reasoning · Grade 6-2 Solids of Revolution and Their Cross-Sections

Problem

Cross-sections cut from a cone

The same cone is cut by a flat plane six different times. Twice square to the axis, high and low; once through the axis; once vertical beside the axis; once slanted right around the side clear of the base; once slanted into the base. Draw each of the six cross-sections.
How it is cut Cross-section How it is cut Cross-section
Your answer
How to solve
Strategy Create a Physical Representation — Imagining six different slices at once is hard, and it is exactly the kind of thing you should not try to do in your head. Carve a cone out of a carrot, or half-fill a cone-shaped cup with water and tilt it, and each cut is there in front of you. Behind the six pictures there is one simple rule to look for: the answer depends only on how the cutting plane sits relative to the axis of rotation, and on whether the plane runs into the flat base. Sorting the six cuts by those two questions turns six problems into three easy families.
1STEP 1

Ask two questions about every plane

For each plane ask about the axis and the base.

2STEP 2

Cuts A and B — planes at right angles to the axis give circles

At right angles to the axis the cut is a circle.

radius of slice ∝ distance from the apex
3STEP 3

Cut C — a plane containing the axis gives an isosceles triangle

Containing the axis gives an isosceles triangle.

two slant sides are equal → isosceles triangle
4STEP 4

Cut D — a vertical plane beside the axis gives a curved dome on a straight base

A vertical plane beside the axis gives a curved dome.

5STEP 5

Cut E — a slanted plane right round the lateral surface gives an ellipse

A slant right round the side gives an ellipse.

6STEP 6

Cut F — a slanted plane that reaches the base gives a half-oval

A slant reaching the base gives a half-oval.

7STEP 7

Check all six against the symmetry rule

All six come out left-right symmetric.

Answer
circle, circle, isosceles triangle, curved dome, ellipse, half-oval
Every cross-section must fit inside the cone, so none of the six can be wider than the base circle, and none is. The two level cuts give circles, and the higher one is the smaller of the two, which agrees with the cone being narrower near the apex. Only the three planes that actually reach the flat base produce a straight edge, and each of those produces exactly one, because a plane meets a plane in exactly one line — the triangle is the exception only because it also passes through the apex, which turns the two lateral edges into straight slant lines as well. Every answer is line-symmetric, as any cut of a spun solid must be. Finally, the largest cut is the one through the axis, and the isosceles triangle is indeed the tallest and widest of the six.
Takeaway

The cut only cares about the axis: straight across gives a circle, right through the tip gives a triangle, tilted gives an oval — and a straight edge appears only when the knife reaches the flat base.

  • Ask two questions about every plane
  • Cuts A and B — planes at right angles to the axis give circles
  • Cut C — a plane containing the axis gives an isosceles triangle
  • Cut D — a vertical plane beside the axis gives a curved dome on a straight base
  • Cut E — a slanted plane right round the lateral surface gives an ellipse
  • Cut F — a slanted plane that reaches the base gives a half-oval
  • Check all six against the symmetry rule