Seventh Grade · Math · week 31

Cross-sections of three-dimensional figures

Identify the two-dimensional shape produced when a plane slices through a three-dimensional solid.

A cross-section is the flat, two-dimensional shape you would see if you sliced straight through a solid and looked at the cut face. The SAME solid can give completely different cross-sections depending on the angle and direction of the slice.

A slice PARALLEL to a solid's base usually matches the base's shape (just possibly smaller, for a pyramid or cone). A slice that cuts straight down through the middle of a curved solid, parallel to its axis, usually reveals a flat shape hiding inside the curve.

📋 For the teacher
Teach it (10–15 min):
  • If you have blocks, clay, or even a loaf of bread at home, actually slice something and look at the cut face -- this idea is much easier to see than to imagine.
  • Ask "parallel to which face?" before predicting any cross-section -- the direction of the slice changes everything.
  • Point out that pyramids and cones shrink their cross-section the closer the slice gets to the tip, while prisms and cylinders stay the same size all the way up.
  • Have them sketch the solid, draw a line showing the slice, and then sketch just the resulting flat shape separately.
Talk about it:
  • Why can the same solid produce different cross-sections?
  • What happens to a pyramid's cross-section as the slice moves closer to its tip?
  • Why does a vertical slice through a cylinder reveal a straight-edged shape instead of a curved one?

Mastery looks like: They correctly predict the two-dimensional shape produced by slicing a three-dimensional solid in a described direction.

Common stumbles: Assuming every slice of a solid matches its base, without considering the direction of the cut; forgetting that a pyramid or cone's cross-section shrinks near the tip.

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Records attendance + 15 min of Math for this learner.
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