3D Shapes — AQA GCSE Mathematics

SPEC G12, G13 GEOMETRY & MEASURES F+H

3D Shapes & Their Views

Faces, edges, vertices for the standard solids — plus how to draw plan, front and side elevations

What you will learn

By the end of this lesson you will:

Define face , edge and vertex with examiner-accepted wording

State the F aces / E dges / V ertices count for the seven standard solids in G12

Verify polyhedron counts using Euler's relation F + V = E + 2

Identify and draw a net of a 3D shape

Construct plan , front elevation and side elevation on squared paper, with hidden edges dashed

Why this matters

3D-shape questions are guaranteed marks across both tiers. Foundation candidates routinely lose marks for the cylinder face count and missing dashed lines. Higher candidates use this content as the foundation for surface area and volume problems (Lessons 4.13–4.14).

Faces, Edges, Vertices

Three terms

Face: a flat (or curved) surface of the solid. A cube has six square faces; a cylinder has two circular faces and one curved face.

Edge: a line where two faces meet. A cube has twelve edges.

Vertex: a point where three or more edges meet (plural: vertices). A cube has eight vertices.

Cube & Cuboid

SolidFacesEdgesVerticesNotes

Cube6128All faces are squares of equal size

Cuboid6128Faces are rectangles (opposite pairs equal)

Cube vs Cuboid

Same F/E/V counts. The difference is shape: a cube has all sides equal; a cuboid has rectangular faces with up to three different edge lengths.

Prisms

Prism

A prism has two identical, parallel faces (the "ends") joined by rectangles. The shape of the ends names the prism.

SolidFacesEdgesVertices

Triangular prism596

Pentagonal prism71510

Hexagonal prism81812

General prism formula

For an n-sided prism: F = n + 2 (the two end faces plus n rectangles); E = 3n (n on each end + n connecting); V = 2n.

Pyramids

Pyramid

A pyramid has a polygon base, with all other faces being triangles meeting at a single point called the apex.

SolidFacesEdgesVertices

Square-based pyramid585

Tetrahedron (triangular-based)464

"Tetrahedron"

A tetrahedron is a triangular-based pyramid — all four faces are triangles. It's the simplest polyhedron.

Curved Solids: Cylinder, Cone, Sphere

SolidFacesEdgesVertices

Cylinder320

Cone211

Sphere100

⚠️ EXAMINER ALERT — cylinder face count. A cylinder has three faces: two circular faces (top and bottom) PLUS the curved surface. Many students answer 2 because they forget the curved surface counts. Same trap on a cone: two faces — the circular base PLUS the curved (slant) surface.

F / E / V Reference Table

Memorise these. They are tested at both Foundation and Higher tier as quick recall questions (1–2 marks each).

SolidFacesEdgesVerticesPolyhedron?

Cube6128Yes

Cuboid6128Yes

Triangular prism596Yes

Square-based pyramid585Yes

Tetrahedron464Yes

Cylinder320No

Cone211No

Sphere100No

Memory aid

For polyhedra (flat-faced solids), F + V = E + 2 always holds — this is Euler's relation (next slide). For curved solids (cylinder, cone, sphere) it does not apply because they have curved faces and curved edges.

Euler's Relation: F + V = E + 2

For any polyhedron (a 3D shape with only flat faces), the number of faces plus vertices is always two more than the number of edges. Use this as a check when you count.

Euler's Relation

F + V = E + 2

SolidFVF + VEE + 2Holds?

Cube68141214✓

Triangular prism5611911✓

Square pyramid5510810✓

Tetrahedron44868✓

⚠️ Curved solids exception. Euler's relation does not apply to cylinders, cones or spheres because they have curved faces. For example, a cylinder: F + V = 3 + 0 = 3, but E + 2 = 4 — they don't match.

Nets

A net is a 2D pattern that folds up to form the 3D shape. The faces of the solid become the regions of the net, joined along the edges that fold together.

Cube nets

A cube has 11 distinct nets (counting rotations and reflections as the same). Every cube net consists of 6 squares arranged so that, when folded along the joining edges, no two faces overlap.

Net of a cylinder

The net of a cylinder is a rectangle (the curved surface unrolled) plus two circles (top and bottom). The rectangle's length equals the circumference of the circles: 2πr.

Plans & Elevations

Three views together describe a 3D shape on flat paper. Each view is a 2D outline drawn on squared paper.

Plan

Looking straight down from above. Shows the shape as if you flew over it.

Front

Looking from directly in front. Width × height as you'd normally see the shape.

Side

Looking from directly to one side. Depth × height.

⚠️ EXAMINER ALERT — hidden lines. Edges that are not visible from that direction must be drawn as dashed lines. Omitting dashes is one of the most common ways to lose marks on plan-and-elevation questions.

Mark scheme conventions

Each view is awarded marks for: (1) the correct outline, (2) correct internal lines showing height changes, and (3) hidden edges dashed. Use the squared paper grid — each unit cube = 1 grid square.

Worked Example 1

F, E, V Identification 3 marks01

A solid has 5 faces, 9 edges and 6 vertices. Verify Euler's relation for this solid and name a possible solid that fits.

M

Substitute into Euler: F + V = E + 2. Then check the F/E/V table for a match.

C

F + V = 5 + 6 = 11. E + 2 = 9 + 2 = 11. The two values are equal, so Euler's relation holds. From the table, F = 5, E = 9, V = 6 matches a triangular prism.

E

Euler's relation holds (11 = 11), and the solid could be a triangular prism. (Note: more than one solid could have these counts — the answer accepts any matching name from the table.)

Worked Example 2: Plan & Elevations

The solid below is made of 6 unit cubes arranged in a staircase: 3 cubes stacked on the left, 2 cubes stacked in the middle, 1 cube on the right. Each cube has side 1 cm.

Sketch the plan, front elevation, and side elevation on a squared grid.

M

Look at the solid from each direction in turn. Draw the silhouette, then add internal lines where heights change. Each unit cube = 1 grid square.

C

See the three views below.

E

Plan: 3 × 1 rectangle (looking down, all three columns project down to a single row of three squares). Front: staircase outline 3 wide at the bottom, going down to 1 wide at the top. Side: 1 × 3 rectangle (looking from the end, only the tallest column is visible).

The Three Views (continued)

Each view of the staircase solid (3 cubes + 2 cubes + 1 cube). Hidden edges that show the column boundaries are drawn as solid lines because they correspond to visible edges of the solid from that direction.

3 squares wide, 1 square deep. Two internal lines mark where heights step.

3 tall on left, 2 in middle, 1 on right.

Just the tallest column — the smaller columns are hidden behind it.

Why the side view is just 1×3

From the side, the 3-cube column at the front blocks the view of the smaller columns behind it. The silhouette is just a 1×3 rectangle. (If we drew it including hidden lines, the 2-cube and 1-cube columns would show as dashed shorter rectangles inside the silhouette.)

Common Errors

Wrong answerCorrect versionWhy

Cylinder has 2 faces3 faces (2 circles + 1 curved)Curved surface counts as a face

Cone has 1 face2 faces (1 circle + 1 curved)Curved surface counts

Sphere has 0 faces1 face (the curved surface)The whole surface is one face

Cylinder has 2 vertices0 verticesNo corners on a cylinder

Confusing F and V counts on a cubeF = 6, V = 8 (different)Faces count surfaces, vertices count corners

Drawing elevations without dashed linesHidden edges shown dashedMarks awarded for showing 3D understanding

Plan view = side viewPlan = looking down ; side = looking acrossThree different views give three different drawings

⚠️ Mark allocation reminder. Each view typically scores: 1 mark for the correct outline shape, 1 mark for correct internal lines (or hidden lines dashed). For a typical "draw plan, front and side" question worth 3 marks, you score 1 mark per view.

Summary

Three terms, eight standard solids, three views

Face : a flat or curved surface. Edge : where two faces meet. Vertex : where edges meet (plural vertices ).

F/E/V counts memorised for: cube, cuboid, triangular prism, square pyramid, tetrahedron, cylinder, cone, sphere .

Euler's relation F + V = E + 2 holds for polyhedra. Use as a count check.

Net : 2D pattern that folds into the 3D shape. A cube has 11 distinct nets.

Three views: plan (top), front elevation , side elevation . Hidden edges dashed .

Common trap

Cylinder has 3 faces, not 2. Cone has 2, not 1. The curved surface always counts as a face.

Top exam tip

Always dash hidden edges in your elevations. It's an easy mark and usually the second mark for each view.

⚠️ Up next: Lessons 4.13 (circles & sectors) and 4.14 (3D mensuration: surface area & volume) build directly on this lesson's vocabulary. Knowing what a "face" means is essential for surface-area calculations.