11 plus 3D Shapes & Nets questions

What a 3D Shapes & Nets question looks like

Solid shapes are one of the geometry topics in an 11 plus maths paper, and the questions count and name rather than measure. A question names or draws a solid and asks how many faces, edges or vertices it has. Or it gives the properties first: which of these five solids has exactly 5 faces? Or it describes a flat net and asks what it folds into. How much a box holds and how far it is round the outside belong to the measurement questions, not to this type.

The solids used are the ones a Year 5 child meets at school: cube, cuboid, sphere, cylinder, cone, triangular, hexagonal and pentagonal prisms, square-based pyramid, and triangular-based pyramid, which some questions call a tetrahedron. Vertices are the corners where edges meet, and the questions use both words.

About half carry a drawing: a solid in outline, a net as shaded squares on a grid, or a stack of centimetre cubes. The three examples below are text-only, where the reasoning is identical.

Some kinds sit outside that pattern. A few print a table of solids with their faces, vertices and edges and ask which row matches a set of numbers, or which row is wrong. A couple ask how many right angles there are altogether on the faces of a cube: six faces, four each, so 24. One stands a stack of centimetre cubes on paper and draws round the bottom, where only the bottom layer counts: a block five cubes long and two deep leaves a 5 by 2 outline with a perimeter of 14 cm.

Example one: the edges of a pentagonal prism

A pentagonal prism has two pentagonal faces. How many edges does it have in total? The options are 12, 10, 15, 17 and 20.

A prism is two matching flat ends joined by straight sides. Count one end: a pentagon has 5 sides, so 5 edges. The other end is the same, another 5. Then the edges joining the two ends, one at each corner of the pentagon, so 5 more. So 5 + 5 + 5 = 15. The answer is 15 edges.

10 is the option to watch. A child who counts both pentagons carefully and stops there lands on it, and it feels like careful work because it was.

Example two: the net of a cylinder

Jake folds a net. The net has 1 rectangle in the centre, with 2 circles attached to the short sides. What 3D shape does it make? The options are cuboid, sphere, cone, triangular prism and cylinder.

Every piece of a net becomes one face. Two circles means two flat round faces, and the rectangle between them rolls up into a tube to join them, with a circle capping each end. The answer is a cylinder.

Cone tempts because a cone does have a flat circle. It has only one, though, and the piece wrapping round it is a fan shape, not a rectangle. Sphere catches a child who reads round and stops: a sphere has no net at all.

Example three: a pyramid sitting on a cube

A solid is made by placing a square-based pyramid on top of a cube, its square base sitting exactly on one face of the cube. How many faces does the combined solid have? The options are 12, 13, 11, 9 and 10.

Count the two solids separately. A cube has 6 faces. A square-based pyramid has 5: the square underneath and 4 triangles leaning up to the point. That is 11 between them. The pyramid’s base is then pressed flat on the top face of the cube, so both of those faces end up inside the solid where nobody can see them. Take away 2, leaving 9. The answer is 9 faces.

11 is the option most children pick, because adding the two totals is all the work they have done. 10 catches the child who takes off one hidden face instead of two: two meet at every join, and both disappear.

How to answer it

1. Read which thing is being counted. Faces are the surfaces, edges the lines where two faces meet, vertices the corners where edges meet. Most wrong options are the right count of the wrong one.
2. For a prism, work from the end face. Edges: the edges of one end, doubled, plus one for each of its corners. Faces: the two ends, plus one side face for every side of the end shape.
3. Count in the same order every time, top then bottom then sides. The faces and edges round the back are the ones that get missed.
4. For a net, count the pieces and name their shapes, because each piece becomes one face. Six equal squares gives a cube, two triangles and three rectangles a triangular prism, one rectangle and two circles a cylinder.
5. If two of the three counts are given, use Euler’s rule: faces + vertices - edges = 2. Put in what you have and work out what you do not.

Common mistakes

• Counting the wrong thing. A cube has 6 faces, 12 edges and 8 vertices, and a question asking for one of them offers the other two as options. Answering 12 for the faces is not a counting slip, it is having drifted onto edges partway through.
• Losing the faces you cannot see. A cube counted as 4 has lost the top and the bottom. A triangular prism counted as 6 edges has missed the three long edges between the ends.
• Trusting a net that looks familiar. One question describes five squares in a cross, one in the middle with a square on each of its four sides, and asks whether it folds into a closed cube. It does not: six faces need six squares, so this net leaves a hole. Counting the squares settles it before any folding is imagined.
• Calling every flat-faced solid a prism. A prism has two matching parallel ends joined by rectangles, which makes a cube and a cuboid prisms too. A square-based pyramid tapers to a point, so it is not one.

How to practise 11 plus 3D Shapes & Nets questions

Most of this is recall. A child who can say without hesitating that a cube has 6 faces, 12 edges and 8 vertices is halfway through every question of this kind before the reasoning starts. Real objects help more than drawings: unfold a cereal box flat, count the pieces, fold it back. Then keep it in rotation, a few at a time: here it is recall going stale that loses the marks. Solid shapes are a small geometry topic in the eleven plus. 11+ Daily has a 3D Shapes & Nets project covering the face, edge and vertex counts, the net matching, the properties tables and Euler’s rule, alongside the other 11 plus maths question types. Every question is original to 11+ Daily, nothing is a past paper, nothing is licensed, and the bank is re-checked in rounds. We have also written about how much 11 plus practice a child needs each day.

Questions parents ask