3D Shapes, Nets, Plans and Elevations
Three-dimensional geometry is about reading a solid accurately from different representations. This lesson develops the language of solids, nets, plans and elevations, and shows why a single view does not always determine a unique object.
Foundation & Higher. The same core content is used on both tiers; more complicated reconstruction problems are useful challenge work rather than a separate Higher-only topic.
Recognising common 3D solids
Use precise geometric features rather than the way a drawing happens to look. A cube has six square faces. A cuboid has six rectangular faces. A prism has a constant cross-section along its length. A pyramid has a base with triangular faces meeting at one apex. Cylinders, cones and spheres have curved surfaces.
For polyhedra, a face is a flat surface, an edge is where two faces meet, and a vertex is a corner. Curved surfaces should not be casually counted as polygonal faces.
Count shared edges and vertices once: a cube or cuboid has 6 faces, 12 edges and 8 vertices; a triangular prism has 5 faces, 9 edges and 6 vertices; a square-based pyramid has 5 faces, 8 edges and 5 vertices. A cylinder has two flat circular ends and one curved surface; a cone has one flat circular base, one curved surface and an apex; a sphere has one curved surface and no edges or vertices.
Prisms and constant cross-sections
A prism keeps the same cross-section all the way along its length. A triangular prism therefore has matching triangular ends. A cylinder also has a constant circular cross-section, but in GCSE terminology it is normally treated separately from polygonal prisms.
A pyramid is different: sections parallel to its base become smaller as they approach the apex.
Nets — thinking about which faces meet
A net is a flat arrangement that folds to make a solid. Having the correct number of faces is not enough: the faces must be arranged so that they fold without overlapping and the correct edges meet.
Worked example: a cuboid has dimensions 5 cm by 3 cm by 2 cm. Its net must contain two 5×3 rectangles, two 5×2 rectangles and two 3×2 rectangles. Matching dimensions tell you which edges can meet.
Plans and elevations
An elevation is an orthographic view: it shows the object straight-on, without perspective. The plan is the view from above. Front and side elevations depend on the stated viewing direction.
Imagine projecting every point straight onto a flat screen. Depth in the viewing direction disappears, so several cubes in a row may appear as one square. That is why the viewing arrow matters.
Reconstructing from several views
A plan tells you which horizontal positions are occupied, while elevations tell you maximum visible heights from particular directions. One view alone can hide depth, and even three views can sometimes allow more than one arrangement.
Worked example: suppose the front elevation has column heights 1, 3, 2 and the plan shows two occupied depth positions behind the middle column. The middle stack must reach height 3 somewhere, but the front view alone cannot tell you whether the height-3 stack is at the front or behind it. Use the plan and side view together before deciding.
Isometric sketches are representations
Isometric paper helps a 3D object look three-dimensional on a flat page. Do not measure an apparent angle or diagonal from the sketch unless the question explicitly tells you the drawing is to scale. Use labelled dimensions and geometric facts.
Think Like a Mathematician
Ask what information a representation has lost. A front elevation loses depth. A plan loses height. A net temporarily loses the 3D connections. Good spatial reasoning comes from combining representations rather than trusting one picture.
Common Mistakes
Calling a cylinder a polygonal prism: a GCSE prism normally has a polygonal constant cross-section; cylinders are treated separately.
Drawing perspective in an elevation: elevations are flat straight-on views.
Ignoring hidden depth: a plan shows occupied positions, not merely the top faces you happen to see in a perspective sketch.
Counting faces only: a supposed net can have six squares and still fail to fold into a cube.
Practice — 3D shapes and views
1. What feature defines a prism?
It has a constant cross-section along its length.
2. How many square faces does a cube have?
6.
3. A triangular prism has how many triangular end faces?
2.
4. A cuboid is 5 cm by 3 cm by 2 cm. List the six rectangular faces needed in its net.
Two 5×3, two 5×2 and two 3×2 rectangles.
5. Which view is seen from directly above?
The plan.
6. Why must a front-elevation question state or show the viewing direction?
Different directions can give different flat views.
7. Can a front elevation by itself always determine the depth of a block arrangement?
No. Blocks can be hidden behind one another.
8. Three stacks of unit cubes stand in one left-to-right row, with heights 1, 3 and 2. Draw the plan, front elevation and side elevation.
The plan is three unit squares in one row. From the front, draw three adjacent columns of heights 1, 3 and 2. From either end of the row, draw one column of height 3: the shorter stacks are hidden. A plan alone would not show the stack heights.
9. Why should you not measure an angle from an isometric sketch?
The sketch represents 3D shape on a flat page and is not necessarily drawn to scale.
10. A solid has a square base and four triangular faces meeting at one point. Name it.
A square-based pyramid.