Math // Geometry

3-D shapes (age 11+)

Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections

Ages
11 to 13
Type
Concept
Builds on
2 topics
Leads to
0 topics
Standards
2 codes

Quick check

If the learner slices a cylinder straight across the middle like cutting a tin of beans in half, can they describe the shape of the cut surface? What about slicing at an angle?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Count and describe the faces, edges, and vertices of a triangular prism and a square-based pyramid
  • Describe the 2-D cross-section produced by slicing a cylinder horizontally or a cone vertically
  • Use properties of 3-D shapes to determine whether a given net will fold into a specified solid

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • 3-D shapes (age 9+)Required

    Geometry // Ages 9 to 10

    KS3 3D shape properties extend KS2 identifying 3D shapes from 2D representations

  • 3-D shapes (age 10+)Required

    Geometry // Ages 10 to 11

    Cross-section visualisation extends KS2 building 3D shapes and making nets

You are here

3-D shapes (age 11+)

Ages 11 to 13 // Concept

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Geometry

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Learners build the idea that a cross-section is a NEW surface created where a slicing plane meets a solid, so its shape depends on the direction of the cut, not just on the solid. Core work is predict-then-slice on real solids (play-doh, carrot, food) at parallel, perpendicular and oblique orientations, naming the cut as distinct from any outer face the solid already had, scaffolding straight cuts before slanted ones, transferring the idea to a new solid such as a cone, counting faces/edges/vertices on the solid handled in the round, and linking a prism's constant cross-section to area-of-base-times-height.

Common misconceptions

  • A solid has one fixed cross-section — a cylinder is 'always a circle' whatever the slice.
  • Confuses the cross-section with a pre-existing outer face, reporting a surface the solid already had.
  • Assumes a 2-D shape extends orthogonally into 3-D, so a slanted cut of a cylinder is judged a circle rather than an ellipse (over-applying 'the slice looks like the base').
  • Counts only the faces and edges visible in a 2-D drawing, missing hidden ones.

Useful representations

  • Sliced play-doh or food solids showing the fresh cut face
  • The cylinder's three cuts: circle (parallel), rectangle (perpendicular), ellipse (oblique)
  • A prism shown as its cross-section swept through a height
  • A face/edge/vertex tally taken from the solid in the round

Teaching practices

  • Physically slice solids at parallel, perpendicular and oblique orientations; predict-then-cut so a naive prediction can be violated.
  • Name the cut as a NEW surface — the shape where the slicing plane meets the solid — distinct from an outer face the solid already had.
  • Scaffold orthogonal cuts before oblique ones: straight slices can be solved by recognising a shape, slanted slices need more precise reasoning (Cohen & Hegarty; Munns et al.).
  • Count faces, edges and vertices on the solid handled in the round, not on the 2-D picture, so hidden elements are included.
  • Treat a prism as a constant cross-section swept along its length, linking to volume as area of base times height.

Activities to try

  1. 1

    Predict, then slice: a cylinder gives more than a circle

    Why it works: A cross-section is the shape where the slicing plane meets the solid, so one solid yields many cross-sections depending on how it is sliced; a cylinder is only 'a circle' when sliced parallel to its base.

    1. Hold a play-doh or carrot cylinder and a plastic knife.
    2. BEFORE cutting, predict AND justify the shape of a straight top-to-bottom slice through the middle; write or say the reason.
    3. Cut and compare the fresh cut face to your prediction (for a true cylinder a rectangle appears, not a circle; a real carrot that tapers gives a near-trapezium, an approximation to the ideal rectangle).
    4. Make a second cut parallel to the circular ends and check that this one does give a circle.
    5. State in your own words why the two cuts give different shapes.
  2. 2

    Tilt the cut: straight before slanted

    Why it works: Assuming a 2-D shape extends straight up into 3-D leads to calling a slanted cut a circle; oblique cuts need more careful reasoning than straight ones, a slanted slice of a cylinder that crosses only the curved surface is an ellipse (one that also cuts an end is an ellipse cut off by a straight edge), and the slice-dependence idea transfers to other solids such as a cone.

    1. After solving the parallel and perpendicular cuts, tilt the knife and cut the cylinder at a slant.
    2. Predict first: is the slanted cut face still a circle? Name the shape you expect.
    3. Cut, then lay the slanted cut face over a traced circle of the same cylinder to compare.
    4. Describe how the oblique cut stretches the circle into an ellipse when the slanted plane crosses only the curved surface. If your cut also runs through a flat end, the face is part of an ellipse with a straight edge, so note which case you got.
    5. Transfer the idea to a new solid: predict then check the cross-section of a cone sliced vertically through its apex (a triangle), to test slice-dependence on a different shape.
  3. 3

    The sweep that makes a prism

    Why it works: A prism is a constant polygonal cross-section swept along its length; this constancy (which a cylinder shares, with a circle as its cross-section) links the solid's properties to volume = area of base x height.

    1. Take a triangular prism (or model one in play-doh).
    2. Slice it parallel to its triangular end, then slice again further along its length.
    3. Compare the cut faces: check whether the cross-section ever changes along the length.
    4. Explain why a polygonal cross-section that never changes (for slices parallel to the end) is what makes the solid a prism, and connect this to volume as area of the end (base) times the length.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
7.G.3
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Geo.15

Math // Geometry

More in Geometry

All 69 topics