Geometry: Nets, plans and elevations of 3D solids

What you'll learn

This topic tests your ability to visualise and represent three-dimensional objects in two dimensions. You'll work with nets (flat patterns that fold into 3D solids), plans (views from above), and elevations (views from the front and side). CIE IGCSE Mathematics papers regularly examine these skills, particularly in Paper 2 and Extended Paper 4, where you must draw accurate representations and identify 3D shapes from their 2D projections.

Key terms and definitions

Net — a two-dimensional pattern that can be folded along edges to form a three-dimensional solid without gaps or overlaps.

Plan — the view of a three-dimensional object as seen from directly above, showing the shape's horizontal cross-section.

Front elevation — the view of a three-dimensional object as seen from directly in front, showing vertical features and height.

Side elevation — the view of a three-dimensional object as seen from the side (usually the right side unless specified), showing depth and height.

Prism — a solid with a uniform cross-section; the plan and one elevation are identical in shape to this cross-section.

Polyhedron — a three-dimensional solid with flat polygonal faces, straight edges and vertices (plural: polyhedra).

Edge — a line segment where two faces of a three-dimensional solid meet.

Face — a flat or curved surface of a three-dimensional solid.

Core concepts

Understanding nets of common 3D solids

A net represents how a 3D solid looks when opened out flat. Different arrangements can form the same solid, so multiple correct nets exist for most shapes.

Cube nets:

• A cube has 11 distinct nets
• Each net must contain exactly 6 square faces
• No more than 4 squares can be in a straight line (this would prevent folding)
• Adjacent squares in the net become adjacent faces in the cube

Cuboid nets:

• Contain 6 rectangular faces (or 2 squares and 4 rectangles if some dimensions match)
• Opposite faces must be congruent rectangles
• Length, width and height dimensions must match correctly when folded

Cylinder nets:

• Two congruent circles (top and bottom)
• One rectangle that wraps around to form the curved surface
• The rectangle's length equals the circumference of the circles: 2πr
• The rectangle's width equals the cylinder's height

Pyramid nets:

• One polygonal base
• Triangular faces meeting at an apex
• For a square-based pyramid: one square base and four congruent isosceles triangles
• The base of each triangle must equal the side length of the square base

Cone nets:

• One circular base
• One sector of a larger circle that forms the curved surface
• The arc length of the sector equals the circumference of the base circle
• The radius of the sector equals the slant height of the cone

Drawing accurate plans and elevations

Plans and elevations are orthogonal projections — views at right angles to the object. Hidden edges are typically shown as dashed lines in technical drawing, though IGCSE questions often omit them for clarity.

Drawing a plan (view from above):

• Position yourself directly above the object
• Draw what you see looking straight down
• Show only visible horizontal surfaces and edges
• Use the correct horizontal dimensions (length and width)
• Ignore height information in the plan

Drawing a front elevation:

• Position yourself directly in front of the object
• Draw what you see looking horizontally at the front face
• Show height and width as they appear from the front
• Include vertical and frontal features
• Depth is not shown in front elevation

Drawing a side elevation:

• Usually the right side unless specified otherwise
• Position yourself directly to the side of the object
• Show height and depth as they appear from the side
• Width is not shown in side elevation
• Curved surfaces appear as their outline from this viewpoint

Key technique for composite solids:

1. Break the object into simpler components (cuboids, cylinders, prisms)
2. Draw each component's plan/elevation separately on scrap paper if needed
3. Combine the outlines, showing the overall external boundary
4. Ensure measurements are consistent across all views
5. Use a ruler for straight edges and compasses for circles

Recognising 3D shapes from 2D representations

Exam questions frequently provide plans and elevations and ask you to identify or sketch the 3D solid.

Matching technique:

• Compare the plan with front and side elevations systematically
• The plan shows the "footprint" — what space the object occupies horizontally
• Elevations show vertical extent and features
• Check that dimensions align: width in plan = width in front elevation; depth in plan = depth in side elevation

Common combinations:

Plan	Front Elevation	Side Elevation	3D Solid
Circle	Rectangle	Rectangle	Cylinder
Square	Square	Square	Cube
Rectangle	Rectangle	Rectangle	Cuboid
Triangle	Rectangle	Triangle	Triangular prism
Circle	Triangle	Triangle	Cone
Square	Triangle	Triangle	Square-based pyramid

L-shaped and composite solids:

• Count the rectangles or sections in each view
• Sketch a rough 3D shape, checking it produces all three given views
• Remember that internal corners might not be visible in all elevations

Working with dimensions and scale

Plans and elevations must be drawn to scale in construction questions. Measurements from these drawings can calculate actual dimensions, surface areas, or volumes.

Scale drawing rules:

• If the scale is 1:50, every 1 cm on the drawing represents 50 cm in reality
• Multiply drawing measurements by the scale factor for actual dimensions
• All three views must use the same scale
• Label dimensions clearly where required

Calculating from plans and elevations:

• A cylinder's volume requires the radius (from plan) and height (from elevation): V = πr²h
• A prism's volume = cross-sectional area × length
• Surface area calculations need dimensions from multiple views
• Always identify which view shows which dimension clearly

Isometric drawing connections

While not strictly plans or elevations, isometric drawings show 3D objects on a 2D surface at specific angles. CIE IGCSE occasionally asks you to relate isometric views to standard orthogonal projections.

Isometric features:

• Drawn on isometric grid paper (equilateral triangle grid)
• Vertical edges remain vertical
• Horizontal edges are drawn at 30° to the horizontal
• All parallel edges in the object remain parallel in the drawing
• Lengths along the three axes remain to scale

Converting between representations:

• An isometric drawing shows three faces (usually top, front, right side)
• Extract the plan by focusing on the top face's shape
• Extract elevations by viewing the appropriate faces straight-on
• Check your projections produce the correct isometric view when reassembled mentally

Worked examples

Example 1: Drawing a net

Question: A closed box is made in the shape of a cuboid measuring 8 cm by 5 cm by 3 cm. Draw an accurate net for this box. [3 marks]

Solution:

Step 1: Identify the faces needed.

• Two faces: 8 cm × 5 cm (top and bottom)
• Two faces: 8 cm × 3 cm (front and back)
• Two faces: 5 cm × 3 cm (left and right sides)

Step 2: Arrange the rectangles so they fold into a box. One possible arrangement (others exist):

        [5×3]
[5×3]   [8×5]   [8×3]   [8×3]
        [8×5]

Step 3: Draw accurately using a ruler.

• Ensure opposite faces have matching dimensions
• Each face connects along a common edge
• Total of 6 faces arranged so folding is possible

Mark allocation: 1 mark for correct dimensions, 1 mark for appropriate arrangement, 1 mark for accurate construction.

Example 2: Identifying a 3D solid

Question: The diagram shows the plan and elevations of a solid object.

Plan: Rectangle 6 cm × 4 cm Front elevation: Rectangle 6 cm × 5 cm
Side elevation: Rectangle 4 cm × 5 cm

(a) Name the solid. [1 mark] (b) Calculate the volume of the solid. [2 marks]

Solution:

(a) All three views are rectangles with consistent dimensions. The solid is a cuboid (or rectangular prism).

(b) Volume of a cuboid = length × width × height

• Length = 6 cm (from plan and front elevation)
• Width = 4 cm (from plan and side elevation)
• Height = 5 cm (from both elevations)

Volume = 6 × 4 × 5 = 120 cm³

Mark allocation: (a) 1 mark for "cuboid"; (b) 1 mark for correct formula or method, 1 mark for correct answer with units.

Example 3: Drawing plan and elevations

Question: A solid is made from 5 identical cubes of edge 2 cm arranged as shown in the isometric drawing:

[Imagine an L-shaped solid: 3 cubes in a row with 2 cubes stacked on the left end]

Draw, full size: (a) the plan [2 marks] (b) the front elevation [2 marks]

Solution:

(a) Plan (view from above): Looking down, you see the footprint of all cubes:

[2×2] [2×2] [2×2]
[2×2]

An L-shape made of four 2 cm × 2 cm squares (the fifth cube is directly below the top-left cube, so invisible from above).

(b) Front elevation (view from front): Looking from the front:

[2×2]
[2×2] [2×2] [2×2]

Shows height: 4 cm on the left (2 cubes stacked), 2 cm for the other two cubes. Width: 6 cm total (three cubes wide).

Mark allocation: 1 mark per view for correct shape, 1 mark per view for accurate dimensions.

Common mistakes and how to avoid them

Mistake: Drawing nets that won't fold properly, with faces that overlap when folded or gaps that appear.

• Correction: After drawing a net, mentally fold it or trace it onto paper to test. Ensure each edge that should join has the same length. For cubes, remember no more than four squares in a straight line.

Mistake: Confusing which dimension appears in which view — particularly mixing up depth and width in elevations.

• Correction: Label a 3D sketch with L (length), W (width), H (height). Plan shows L and W; front elevation shows L and H; side elevation shows W and H. Check measurements are consistent across views.

Mistake: Drawing a plan that shows features visible from the side or front, essentially drawing a 3D representation instead of a true overhead view.

• Correction: Plans show only the horizontal outline. Imagine the object pressed flat under glass from above — draw only what touches the glass. Ignore all height information.

Mistake: For cylinders, drawing the rectangle in the net with incorrect dimensions, forgetting that its length must equal the circumference of the circular bases.

• Correction: Calculate the circumference first: C = 2πr or πd. The rectangle's longer side must equal this value exactly, and the shorter side equals the cylinder's height.

Mistake: In composite solids, drawing elevations that don't match the plan, particularly when sections are hidden behind others.

• Correction: Work systematically. On the plan, number or label different sections. For each section, determine its height from elevations. Build the 3D shape mentally or with blocks if possible.

Mistake: Missing hidden detail in elevations, such as internal corners or steps, resulting in incomplete or inaccurate drawings.

• Correction: Trace the outline carefully. Where the plan shows a change in depth (L-shapes, steps), the elevation must show a change in height or width. Every feature in the plan creates a corresponding feature in at least one elevation.

Exam technique for Geometry: Nets, plans and elevations of 3D solids

Command word awareness:

• "Draw" requires accurate construction with ruler and compasses. Marks for precision and correct dimensions.
• "Sketch" allows freehand but must show correct shape and proportions. Fewer marks but still must be clear.
• "Name" or "Identify" the solid requires the specific geometric term (cube, cuboid, cylinder, prism, pyramid, cone).
• "On the grid" means use the squares provided; each square usually represents a specific measurement given in the question.

Marks allocation patterns:

• Nets: typically 2-3 marks (1 for correct faces/shapes, 1-2 for arrangement and accuracy)
• Single plan or elevation: 1-2 marks (1 for shape, 1 for dimensions)
• Both plan and elevations together: 3-4 marks total
• Identifying solids: 1 mark for correct name

Time-saving strategies:

• Questions asking for both plan and elevations: draw the plan first, as it often helps visualise the other views
• Use pencil and keep a ruler handy throughout the geometry section
• Grid paper questions: count squares carefully before drawing
• If a net seems wrong, check opposite faces match before redrawing everything

Presentation matters:

• Use a sharp pencil for clean lines
• Label dimensions where required or if it clarifies your answer
• For construction marks, ensure lines meet precisely at corners
• If redrawing, erase thoroughly to avoid confusion

Quick revision summary

Nets are 2D patterns folding into 3D solids; check dimensions match on joining edges. Plans show overhead views (length × width), front elevations show front views (length × height), side elevations show side views (width × height). Match dimensions across all views to identify solids: rectangles suggest cuboids/prisms, circles indicate cylinders/cones. For cylinders, net rectangle length = 2πr. Always draw accurately with ruler, check foldability for nets, and ensure projections show only visible outlines from each specified direction. Practice mental rotation to visualise 3D shapes from 2D representations.