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Three dimensional shapes nets and spatial reasoning - Mathematics Year 6

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Introduction

Six squares do not automatically make a cube net. Learn to recognise complete solids, track faces while folding and explain why an attractive-looking arrangement can still overlap.

Success criteria. Describe solids and their nets using faces, edges and vertices; select a reliable representation for an unfamiliar spatial problem; and communicate a checked solution with labels and stated assumptions.

Three chapters provide original explanations, unchanged Commons diagrams, an original nets video and native activities. The labelled letter layouts are newly authored conceptual square grids, separate from the Commons artwork. They model ideal hinged faces, without extra cuts, stretching or overlapping; online reasoning does not certify practical cutting or construction skill.

Describe the entire solid rather than only its visible faces

Three kinds of features. In a polyhedron, a face is a flat polygonal region, an edge is a line segment shared by faces, and a vertex is a corner where edges meet. Count each feature once, including hidden features.

Cube and cuboid. A cube has six equal square faces, twelve edges and eight vertices. A cuboid has six rectangular faces, twelve edges and eight vertices; a cube is a special cuboid whose dimensions are equal. Not every cuboid is a cube. The same feature counts do not settle whether every face is square.

Build the counts. For a cube, six faces each have four edges, giving twenty-four face-edge incidences. Every edge belongs to two faces, so there are twelve distinct edges. Six faces each have four corners, giving twenty-four face-corner incidences; three faces meet at each vertex, so there are eight distinct vertices.

$6 \times 4 \div 2 = 12, 6 \times 4 \div 3 = 8$

These divisions use the cube's incidence relationships. Do not apply them blindly to every different solid.

Triangular prism. Two congruent triangular ends are joined by three lateral faces. The complete prism has five faces, nine edges and six vertices. A right triangular prism has rectangular lateral faces; an oblique prism has parallelogram lateral faces. The counts remain five, nine and six.

Read the actual source model. Its triangular end at the top is visible, while the other triangular end and some connecting features are hidden. The intended mathematical vertices are triangle corners, not extra corners around the decorative physical joints. Two triangle boundaries supply six edges, with three corresponding joining edges:

$2 \times 3 + 3 = 9, 2 \times 3 = 6$

The second result counts the three vertices at each of two ends. Perspective does not remove hidden edges or create new mathematical ones.

A square-based pyramid for contrast. One square base and four triangular sides make five faces. Four base edges plus four edges to the apex make eight edges; four base corners plus the apex make five vertices. This distinguishes it from a triangular prism even though each has five faces.

Ideal solid	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Triangular prism	5	9	6
Square-based pyramid	5	8	5

Curved surfaces need their own description. A sphere, cylinder and cone have curved surfaces, so they are not polyhedra. Describe the cylinder's two flat circular ends and curved side, or the cone's circular base, curved side and tip, rather than forcing them into this polygon-face table.

Independent attempt. Describe a cube, cuboid and triangular prism completely. Give face shapes as well as counts, explain the hidden features of the source prism and show why counts alone cannot distinguish a cube from every cuboid.

Test a net for connected faces, closure and overlap

A net is an unfolding of a solid's surface. Its faces meet along fold edges. It must fold to cover the intended surface once, with matching edges and no overlaps or missing faces. Glue tabs help assembly but are not additional faces of the solid.

Six squares are necessary but not sufficient for a cube net. Face count, congruent square shape and edge connection are initial checks. You must also track where every face goes when folded.

Keep the source and our models distinct. The source surveys eleven cube nets. The letter layouts below are newly authored models: every letter represents one congruent square cell, and a blank is no face. Adjacent letter positions share a full edge in the conceptual grid. Letter spacing is a schematic, not a physical measurement.

Net P — a valid cross.

  A
B C D
  E
  F

Use C as a reference face. B and D fold to opposite side faces, A and E to another opposite pair, and F folds opposite C. All six faces occupy distinct cube faces, and their edges close. The extra square F is not a second bottom face: it folds once more around E.

Net S — a second valid arrangement.

  A
B C D E
  F

The four-square row can wrap around four side faces; A and F close the remaining pair. Relative to reference C, the opposite pairs are A/F, B/D and C/E. A different flat arrangement can make the same kind of solid.

Net Q — six squares that fail.

A B
C D
E F

This two-by-three rectangle has the right number of equal squares and is connected, but it is not a cube net. At its internal four-square meeting points, four face corners remain joined. A cube vertex joins three face corners. Folding the rigid squares into cube faces would create incompatible folds or overlap unless extra cuts were introduced.

Net R — a strip that fails.

A B C D E F

Wrapping the strip around four side faces brings later squares onto already occupied faces and leaves the other pair of faces uncovered. Six squares in a row therefore do not form a closed cube.

Corbettmaths' original Nets - Corbettmaths connects flat arrangements with solids. Compare its cube example with our P and S layouts. Predict a fold before viewing the explanation; the written route supplies the complete reasoning and feedback independently.

A reusable test sequence. Identify the intended solid; check face shapes, sizes and count; check full-edge connection; choose one reference face; track each neighbour and further fold; then test that every required face is covered once and all boundary edges match.

Independent attempt. Classify P, S, Q and R before reading the assessment. For each valid model, give the opposite face pairs. For each invalid model, explain a specific conflict rather than only say it looks wrong. State that no extra cuts, stretching or face overlap are allowed.

Track opposite faces and justify an unfamiliar spatial conclusion

Flat distance is not the same as final adjacency. In P, C and F are separated by E in the flat layout, but become opposite cube faces. A and E are also opposite; neither shares a cube edge with the other. Two squares that touch along a fold edge in the net do remain adjacent faces after folding.

A viewpoint can change without changing the object. Rotating or reflecting a complete net on paper does not by itself make a valid cube net invalid. Rotating the folded cube changes which face looks front-facing, but its opposite-pair relationships remain. State the labels rather than rely on the words left or top without a reference view.

Check a cuboid net with dimensions. Consider an ideal 3 cm by 2 cm by 1 cm cuboid. Its six faces form three congruent opposite pairs: two 3-by-2 rectangles, two 3-by-1 rectangles and two 2-by-1 rectangles. Merely collecting six arbitrary rectangles is insufficient: matching fold edges need the correct lengths.

The total flat face area for these ideal dimensions is:

$2 \times (3 \times 2 + 3 \times 1 + 2 \times 1) = 22$

That is 22 cm² of surface, while the enclosed volume is 6 cm³. This is an extension connecting a net's coverage with the earlier area/volume course, not a claim that square and cubic units are interchangeable.

A right triangular-prism net. It needs two congruent triangular ends and three rectangles whose widths match the three triangle side lengths, all with a common prism length. A circle does not replace a triangular end, and three triangles plus two rectangles have the wrong face set for this solid.

A useful count check. For the listed closed convex polyhedra, faces minus edges plus vertices equals two:

$6 - 12 + 8 = 2, 5 - 9 + 6 = 2, 5 - 8 + 5 = 2$

This supports a consistency check of the cube/cuboid, triangular prism and square-based pyramid counts. It does not prove that an arbitrary six-square arrangement is a valid net: geometric closure and overlap still have to be checked.

A complete unfamiliar argument. Suppose someone proposes an arrangement with two triangles and three rectangles. That face inventory is consistent with a right triangular prism, but alone is not enough to guarantee a net. Check congruent ends, matching rectangle widths, a shared prism length and a fold arrangement that closes without overlap.

Coordinate descriptions make a model reproducible. In our P layout, count columns from zero left to right and rows from zero downwards. The face positions are A(1,0), B(0,1), C(1,1), D(2,1), E(1,2) and F(1,3). A translation or rotation of this whole coordinate arrangement changes its location, not the intended edge connections.

Tasks for your spatial dossier. Describe the cube, cuboid and triangular prism; compare the pyramid; classify all four letter nets; give the opposite pairs for P and S; and assess the 3-by-2-by-1 cuboid's face inventory. Use source diagrams as evidence about complete ideal solids, not as photographs from which numerical dimensions are guessed.

Repair these claims. Six connected squares always make a cube; a visible prism image has only the faces you can see; the source's number eleven means eleven faces; equal face counts prove two solids have identical shapes. Give a precise counterexample or corrected description for each.

Communicate your check. Name the ideal solid and assumptions, describe faces/edges/vertices accurately, use a reproducible layout or fold trace, explain closure and identify one independent consistency check. State what a count check proves and what it leaves unresolved. Actual safe tool handling or physical construction needs separate observation.

Interactive Tasks

Quiz: Test Your Knowledge

Which cube feature counts are correct? (Six faces, twelve edges and eight vertices) (!Six faces, eight edges and twelve vertices) (!Twelve faces, six edges and eight vertices) (!Only the three faces visible in a typical drawing)

Why is every cube a cuboid under this course’s definitions? (Its square faces are special rectangles) (!A cuboid must have no equal dimensions) (!Every solid has six faces) (!Perspective makes all faces equal)

How many faces, edges and vertices does a triangular prism have? (Five faces, nine edges and six vertices) (!Five faces, eight edges and five vertices) (!Six faces, twelve edges and eight vertices) (!Only one triangular face and three vertices)

What do the source picture’s numbers one to eleven identify? (Eleven different whole nets) (!The number of faces in each solid) (!Eleven corners of one cube) (!The number of extra glue tabs required)

Which of our four square arrangements are valid cube nets? (P and S) (!Q and R) (!Every connected arrangement of six squares) (!Only the straight strip R)

Which face is opposite C in P? (F) (!A) (!B) (!E)

Which face is opposite A in S? (F) (!C) (!D) (!E)

Why does the two-by-three rectangle Q fail as a cube net? (Its joined faces impose incompatible folds or overlap rather than six distinct closing cube faces) (!It has fewer than six squares) (!Its squares are all different sizes by definition) (!Its rows are disconnected)

Which face set belongs to a right triangular prism? (Two congruent triangles and three appropriately matched rectangles) (!Three triangles and two rectangles) (!Two circles and one rectangle) (!Six arbitrary squares)

What is the limit of a faces-minus-edges-plus-vertices count check? (It can check stated counts but cannot alone prove a net closes without overlap) (!It proves any flat arrangement is valid) (!It counts only visible faces) (!It makes dimensions and edge connections unnecessary)

Memory Game

Face	A flat polygonal region of a polyhedron
Edge	A line segment shared by faces
Vertex	A corner where edges meet
Net	A flat unfolding of a solid’s surface
Congruent	The same shape and size
Opposite faces	Faces across a solid rather than sharing an edge
Closure	Covering the intended complete surface without gaps or overlaps

Drag and Drop

Count solid features	Include hidden ideal faces, shared edges and vertices once each
Check a cube face inventory	Require six congruent square faces before testing the arrangement
Trace a net	Choose a reference face and follow every full-edge fold
Reject an overlapping arrangement	Identify the repeated face or incompatible fold rather than rely on appearance
Check a cuboid net	Match each rectangle’s dimensions and its corresponding joining edge

Crossword Puzzle

VERTEX	A corner of a polyhedron
EDGES	Line segments shared by faces
FACES	Flat polygonal regions bounding a polyhedron
CUBE	A solid with six equal square faces
PRISM	A solid with matching parallel polygonal ends
FOLD	Turn a face around a shared hinge edge

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Three solid descriptions: Describe cube, cuboid and triangular prism with face shapes and correct feature counts. Include hidden features and the cube/cuboid relationship.
Four net decisions: Classify P, S, Q and R under the stated no-extra-cut assumptions. Give a reason for each.
Source interpretation: Explain what the eleven printed source numbers label and why physical corner fittings in the prism drawing do not create extra mathematical vertices.
Count reconstruction: Derive the cube edge/vertex counts from face incidences and the triangular-prism counts from its two ends and joining features.

Standard

Opposite pairs in P: Give all three opposite face pairs for P. Explain how F folds around E and why C/F do not share a cube edge.
Opposite pairs in S: Give all three opposite face pairs for S. Compare its valid arrangement with P without confusing paper position and solid adjacency.
Cuboid face inventory: Describe the three opposite rectangle pairs of a 3 cm by 2 cm by 1 cm cuboid. Explain why arbitrary rectangles with the right count are insufficient.
Prism versus pyramid: Compare the triangular prism and square-based pyramid. Explain why both can have five faces while their edge/vertex counts and face shapes differ.

Advanced

Two invalid-net arguments: Explain the four-square corner conflict in Q and the repeated side-face problem in R. State why extra cutting would change the task.
Transformation reasoning: Translate, rotate or reflect the full P layout in a sketch. Explain which connections and opposite pairs must remain, without claiming a new orientation creates a different solid type.
Inventory versus sufficiency: Assess a proposed two-triangle three-rectangle net. Explain the dimension and closure checks needed before claiming it forms a right triangular prism.
Complete spatial dossier: Create or assess an unfamiliar net. Provide a reproducible layout, face inventory, fold/closure argument and an independent count check; distinguish mathematical evidence from actual construction competence.

Learning Check

Solid feature counts: Check the three requested solids and pyramid comparison.
Four net decisions: Check P, S, Q and R.
Opposite pairs: Check every pair in each valid letter net.
Cuboid and prism inventories: Check dimensions and sufficiency.
Incidences and consistency: Check the independent count arguments.
Complete spatial explanation: Assess a complete original argument.

Worked checks and assessment guidance

Cube: 6 equal square faces, 12 edges, 8 vertices. Cuboid: 6 rectangular faces, 12 edges, 8 vertices, with the cube a special case. Triangular prism: 2 congruent triangular ends and 3 lateral faces, giving 5 faces, 9 edges, 6 vertices; right-prism lateral faces are rectangles. Square-based pyramid: 5 faces, 8 edges, 5 vertices.
P and S are valid cube nets. Q is invalid: the two-by-three arrangement imposes incompatible folds at joined four-square points or overlap. R is invalid: a six-square strip repeats side faces rather than supplying six distinct closing faces. Six connected congruent squares alone are insufficient.
P: A/E, B/D, C/F. S: A/F, B/D, C/E. Choose a reference face and track the folds; flat distance is not a reliable substitute. The letters belong to the newly authored models, not to the Commons image’s net numbers.
For the 3-by-2-by-1 cm cuboid, opposite pairs are two 3×2, two 3×1 and two 2×1 cm rectangles. Their total area is 22 cm², while volume is 6 cm³. A right triangular prism needs two congruent triangular ends and three rectangles matching the triangle sides and common prism length. Face inventory alone does not prove closure.
Cube face-edge incidences: 6×4 divided by 2 gives 12 edges. Cube face-corner incidences: 6×4 divided by 3 gives 8 vertices. Triangular prism has 2×3+3 = 9 edges and 2×3 = 6 vertices. For the listed closed convex polyhedra, faces−edges+vertices = 2; this is not a sufficient net test.
Require named solid, ideal-face assumptions, correct counts and shapes, reproducible full-edge layout, matching dimensions where needed, a fold/closure/overlap explanation and a distinct consistency check. Include hidden features, distinguish source net numbers from face labels and avoid claiming a physical construction or measuring skill from online work.

Learning evidence

Keep complete descriptions of three solids, all four net decisions, six labelled opposite pairs across P and S and a checked unfamiliar net argument. Include hidden features, a reproducible layout, closure evidence and a separate count check. Physical cutting, folding and safe tool use require separate observation.

OER and Wikipedia

Checked 11 November 2026. Department for Education: mathematics programmes of study, Year 6 geometry, informs describing three-dimensional shapes and their nets, accurate feature vocabulary and spatial reasoning.

Original P/S/Q/R letter layouts, fold arguments, tasks and feedback are newly authored. Their equal-cell coordinates are checked independently using right-angle face orientations, matching unit-cube edges and closed-surface feature counts. The course does not claim that a feature-count identity alone proves a net valid.

Embedded original Nets - Corbettmaths, corbettmaths. The original is embedded rather than copied or described as Commons-licensed. Written explanations and native practice provide an independent route.

Commons assets are unchanged. Source numbers 1–11 identify whole cube nets, not face labels. The prism’s hidden faces remain part of the ideal solid; physical joint details are not extra mathematical vertices. Right triangular-prism rectangle tasks are distinguished from general oblique prisms.

Triangular prism.svg, Triangular_prism.png: mathematics derivative work: Lovasoa (talk), CC BY-SA 3.0. Unchanged.

Cube net pic.png, BergerGießen, CC BY-SA 4.0. Unchanged.