1MA1 · G12–G13Geometry & MeasuresFoundation & HigherGrades 2–4

Plans & Elevations

Architects and engineers describe 3D objects with flat 2D drawings: a $\textbf{plan}$ shows the view from directly above, and $\textbf{elevations}$ show the views from the front and the side. Being able to move between a solid and its views — in both directions — is a visual skill that Edexcel tests on isometric grids and squared paper.

30 min Mensuration

The big picture

Each view is a projection: imagine looking straight at the object from one direction and drawing only the outline and edges you would see, with no perspective. A cuboid looks like a rectangle from every direction; a cylinder looks like a circle from above and a rectangle from the side.

Questions range from naming faces, edges and vertices, through drawing the plan and elevations of a solid made of cubes, to working out a solid from its views and finding its volume.

What you'll learn

Describe 3D shapes using faces, edges and vertices
Draw solids on isometric paper
Draw plans, front elevations and side elevations
Interpret plans and elevations to identify a solid or find its volume

Properties of 3D shapes

A $\textbf{face}$ is a flat surface, an $\textbf{edge}$ is where two faces meet, and a $\textbf{vertex}$ is a corner. A cube has 6 faces, 12 edges and 8 vertices.

A $\textbf{prism}$ has a constant cross-section: a triangular prism has 5 faces, 9 edges and 6 vertices. A $\textbf{pyramid}$ has a base and triangular faces meeting at an apex: a square-based pyramid has 5 faces, 8 edges and 5 vertices.

For polyhedra (solids with flat faces), Euler’s formula links them: faces $+$ vertices $-$ edges $= 2$ .

F + V - E = 2

Euler’s formula for polyhedra.

Your turnSame method, new numbers — try it before moving on.

How many edges does a triangular prism have?

Your answer

Show worked solution

Solution

1Two triangles: $3 + 3 = 6$ edges.
2Three edges join them: $6 + 3 = 9$ .

Answer: $9$

Isometric drawings

Isometric paper has dots in a triangular pattern, so vertical edges are drawn vertically and horizontal edges run at $30^{\circ}$ to the horizontal.

Lengths along the grid lines are true lengths, so a cube of side 2 has edges two dot-spaces long.

Draw visible edges only, and keep the orientation consistent.

Tip — Start isometric drawings from the nearest vertical edge and build outwards. It keeps the shape from skewing.

Plans and elevations

The $\textbf{plan}$ is the view from directly above. The $\textbf{front elevation}$ is the view from the front, and the $\textbf{side elevation}$ is the view from the side (the question will show which side with an arrow).

Draw each view to scale, showing the full outline. Draw internal lines where there is a change in depth or height.

Heights appear in elevations; the plan shows only length and width.

Example 1A solid is made from centimetre cubes: a bottom layer of 3 cubes in a row, with 1 cube stacked on the left-hand end. Describe its plan, front elevation and side elevation (from the right).

1Plan: a

3 \times 1

rectangle (from above you see three squares).

2Front elevation: an L-shape — three squares along the bottom with one on top at the left.

3Side elevation from the right: a

1 \times 2

rectangle, since the solid is 1 cube deep and 2 cubes tall at its highest.

AnswerPlan

3 \times 1

rectangle; front view L-shape; side view

1 \times 2

rectangle.

Why this works

The side view hides detail: from the right you cannot tell that the stacked cube is at the far end. That is why a single view is rarely enough to identify a solid, and why engineers draw three.

From views to solids

Combine the plan (footprint) with the elevations (heights) to reconstruct the solid.

Once the solid is known, find its volume or surface area as usual.

Example 1The plan of a solid is a 4 cm by 3 cm rectangle. Its front and side elevations are both rectangles of height 2 cm. Name the solid and find its volume.

1A rectangular footprint with a constant rectangular height is a cuboid.

2Volume

= 4 \times 3 \times 2

AnswerA cuboid with volume 24 cm³

Your turnSame method, new numbers — try it before moving on.

The plan of a solid is a circle and its front elevation is a rectangle. Name the solid.

Your answer

Show worked solution

Solution

1From above: a circle. From the front: a rectangle.
2The solid is a cylinder.

Answer: cylinder

Think like an examiner

Draw views to the scale given, using a ruler.
Include internal lines where heights or depths change.
Label each view as plan, front elevation or side elevation.

3D shapes

F + V - E = 2

Euler’s formula.

\text{plan: view from above}

Plan.

\text{elevation: view from front or side}

Elevations.

Watch out for these

Drawing a 3D perspective view instead of a flat elevation.

Elevations are flat 2D outlines.

Showing height on the plan.

The plan shows only the footprint.

Miscounting a cube’s edges as 8.

A cube has 12 edges and 8 vertices.

Drawing the side elevation from the wrong side.

Follow the arrow in the question.

Stretch yourself

A prism has a cross-section that is a regular hexagon. How many faces, edges and vertices does it have? Check your answer with Euler’s formula.

Hint — A prism has two copies of the cross-section joined by rectangles, one for each side of the cross-section.

Reveal solution

SolutionFaces: 2 hexagons

+ 6

rectangles

= 8

. Edges: 6 on each hexagon plus 6 joining edges

= 18

. Vertices: 6 on each hexagon

= 12

. Euler:

F + V - E = 8 + 12 - 18 = 2

✓.

\blacksquare

Questions students ask

What is the difference between a plan and an elevation?

A plan is the view from above; an elevation is the view from the front or side.

Do I need to draw hidden edges?

Not in plans and elevations at GCSE — draw the visible outline and edges.

What is isometric paper for?

Drawing 3D shapes so that vertical lines stay vertical and lengths along grid lines are accurate.

Key takeaways

Faces, edges and vertices describe 3D solids; $F + V - E = 2$ .
Isometric drawings keep vertical edges vertical and true lengths.
Plan: from above. Front and side elevations: from the front and side.
Combine views to reconstruct a solid.
Then find volume or surface area as normal.

Exam-style questions

Practise it the way it is examined

3 original questions · 5 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

Write down the number of faces, edges and vertices of a square-based pyramid.[2]
Show mark scheme
- B1Two correct
- B15 faces, 8 edges, 5 vertices

Question 2Grade 31 mark

A solid is made from centimetre cubes. Its plan is a 3 cm by 2 cm rectangle, and its front and side elevations are rectangles of height 2 cm.

How many cubes are used?[1]
Show mark scheme
- B1 $12$
Answer: $12$

Question 3Grade 42 marks

A prism has 12 vertices. How many faces and edges does it have?[2]
Show mark scheme
- M1Recognises a hexagonal prism
- A18 faces and 18 edges

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

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