Plans and elevations
Interpreting the plan, front elevation and side elevation of a 3D solid, and counting cubes and view squares
About two lessons of 45 to 60 minutes
Architects describe a whole building with three flat drawings
No architect hands a builder a 3D sculpture of a house. They hand over flat drawings: the PLAN (the outline seen from directly above, the view a drone camera pointing straight down would capture), the FRONT ELEVATION (the flat view walking up to the front door), and the SIDE ELEVATION (the flat view from the side). Three 2D pictures, together pinning down one 3D object.
GCSE asks you to move both ways: given a solid, describe or draw its three views; given views, reason about the solid. For solids built from centimetre cubes every question becomes counting: the plan counts occupied positions, and each elevation shows every column of the view at the height of its TALLEST stack, because a taller stack hides everything behind it.
- A house plan on a property listingthe plan view: walls seen from directly above, no heights at all
- A skyline photographan elevation: you see each building's width and height, but not its depth
- A LEGO model checked against its instructionsinstruction booklets show plans and elevations of each build stage
- A tin of beans on the shelfplan: a circle. Front elevation: a rectangle. Two views, and the tin's shape is fixed
What students will be able to do
Students will interpret the plan, front elevation and side elevation of a 3D solid, name the 2D shape each view of a standard solid makes, and count the cubes in a unit-cube solid and the squares in each of its three views.
- I can say which direction each view looks from: plan from above, front elevation from the front, side elevation from the side.
- I can name the 2D shape of each view of a cylinder, cone, sphere, pyramid, prism, cube or cuboid in a stated position.
- I can give the dimensions of a cuboid's plan (width by depth), front elevation (width by height) and side elevation (depth by height).
- I can count the cubes in an isometric drawing of a cube solid when told none are hidden.
- I can count the squares in each view of a cube solid: occupied positions for the plan, tallest stack per column for an elevation.
Standards this unit teaches
- GCSE Geometry and measures #13UK GCSE Mathematics (DfE, England)Plans and elevations of 3D shapes
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Geometry and measures" section, "Properties and constructions", item 13, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "construct and interpret plans and elevations of 3D shapes." Standard type: taught and assessed for every GCSE student, Foundation and Higher.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Plan
- the 2D view of a solid seen from directly above
- Front elevation
- the 2D view of a solid seen from the front, showing width and height
- Side elevation
- the 2D view of a solid seen from the side, showing depth and height
- Isometric drawing
- the standard way of drawing a 3D cube solid on paper, with vertical edges drawn vertical and horizontal edges at 30 degrees
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. The three views of a solid
ConcreteOne solid, three cameras: directly above (plan), straight ahead (front elevation), from the side (side elevation). Each camera flattens one dimension away: the plan loses height, the front elevation loses depth, the side elevation loses width. The figure shows a cube solid with all three of its views derived from the same arrangement.
Build the figure's solid from linking cubes if you have them, and let students check each drawn view by closing one eye and sighting along each camera direction in turn.
- Which of the three views can never show the solid's height? Why?
- Two different solids can share the same plan. Can you describe a pair?
2. Views of the everyday solids
PictorialEach standard solid has views worth knowing cold, but they depend on how the solid is RESTING, so exam questions always state the position. A cylinder standing on its circular base: plan a circle, both elevations rectangles. A cone on its base: plan a circle with a dot at its centre (the apex from above), elevations triangles. A sphere: circles from everywhere. A square-based pyramid on its base: plan a square with both diagonals (the four sloping edges from above), elevations triangles.
A cuboid is 6 cm wide, 4 cm deep and 2 cm tall. Give the dimensions of its plan, front elevation and side elevation.
- The plan looks down on the top face, so it shows width by depth: 6 cm by 4 cm.
- The front elevation shows width by height: 6 cm by 2 cm.
- The side elevation shows depth by height: 4 cm by 2 cm.
Answer: Plan 6 cm by 4 cm; front elevation 6 cm by 2 cm; side elevation 4 cm by 2 cm.
- Why does a cylinder lying on its curved surface have a DIFFERENT plan from one standing on its base?
- Which solid has the same view from every direction, and what is that view?
3. Counting cubes and view squares
AbstractFor a solid built from centimetre cubes, every view question becomes a count. Cubes: add up every stack. Plan squares: one per occupied position. Elevation squares: each column of the view shows only its TALLEST stack, because taller stacks hide shorter ones behind them; add the per-column maximums.
The figure's solid is built from centimetre cubes (none hidden). Find (a) the number of cubes, (b) the plan squares, (c) the front elevation squares, and (d) the side elevation squares.
- (a) Add every stack: back row 3 + 2 + 1, front row 1 + 1, total 8 cubes.
- (b) Plan: count occupied positions: 3 in the back row + 2 in the front row = 5 squares.
- (c) Front elevation: tallest stack in each column: max(3,1) + max(2,1) + max(1,0) = 3 + 2 + 1 = 6 squares.
- (d) Side elevation: tallest stack in each row: front row max 1, back row max 3, so 1 + 3 = 4 squares.
Answer: (a) 8 cubes (b) 5 squares (c) 6 squares (d) 4 squares
- Why does the front elevation use the TALLEST stack in each column instead of adding both rows' stacks?
- Can the plan ever have MORE squares than the solid has cubes? Why not?
Common misconceptions and how to address them
MisconceptionDrawing a 3D picture (an isometric sketch) when asked for a plan or elevation.
Why it happens: The solid is presented in 3D, so students reproduce what they see instead of flattening it to the requested viewpoint.
How to address it: Repeat the rule 'a view is FLAT': every plan and elevation is a 2D outline with no slanted lines and no depth. If a drawing shows more than one face direction of the solid, it is not a view.
MisconceptionAdding EVERY stack's height into an elevation, instead of taking the tallest stack per column of the view.
Why it happens: Counting all the cubes works for the total, so students reuse the same all-of-them habit for elevations.
How to address it: Physically line up two stacks one behind the other and look from the front: the shorter one disappears. Each column of an elevation shows exactly the tallest stack in that line of sight, so elevations add MAXIMUMS, never totals.
MisconceptionConfusing which two dimensions each view shows for a cuboid.
Why it happens: Width, depth and height all look interchangeable on a drawn cuboid.
How to address it: Anchor each view to the dimension it destroys: the plan looks DOWN so it loses height (leaving width by depth); the front elevation looks IN so it loses depth (leaving width by height); the side elevation loses width (leaving depth by height).
Guided practice (with answers)
1. A solid is one flat layer of cubes covering a 5 by 3 rectangle. How many cubes, and how many plan squares?
Answer: 15 cubes and 15 plan squares: one layer means every cube shows from above.
2. A cuboid tower is 3 cubes wide, 2 deep and 4 tall. How many cubes, and how many squares in its front elevation?
Answer: 3 x 2 x 4 = 24 cubes; the front elevation is a 3 by 4 rectangle of 12 squares.
3. What is the plan view of a cylinder standing on its circular base?
Answer: A circle: looking straight down, only the circular top shows.
4. What is the front elevation of a cone standing on its circular base?
Answer: A triangle: the sloping surface flattens to two straight sides meeting at the apex.
5. What is the plan view of a square-based pyramid standing on its base?
Answer: A square with both diagonals drawn: the four sloping edges seen from directly above.
Independent practice worksheets
Practise all three views, on cube solids and on the everyday solids, with computed, never-wrong answer keys and code-drawn isometric figures.
Differentiation
- Build every early example from real linking cubes and photograph it from above, the front and the side; match each photo to its drawn view.
- Give the three-camera sentence with every question: plan = bird, front = person walking up, side = person at the side window.
- For elevations, colour the tallest stack in each column of the view before counting, to make the 'maximums not totals' rule visible.
- Given a plan, a front elevation and a side elevation, build (or sketch) a cube solid that matches all three, then decide whether the answer is unique.
- Find two DIFFERENT cube solids sharing the same three views, and explain what extra information would tell them apart.
- Draw the plan and elevations of a cylinder lying on its curved surface, and compare them with the standing cylinder's views.
Assessment: exit ticket
A three-question exit ticket: one cube count, one everyday solid, one cuboid view size.
1. A cube solid is 2 cubes wide, 1 deep and 3 tall. How many squares are in its front elevation?
Answer: 6 squares: the front elevation is a 2 by 3 rectangle.
2. Name the plan, front elevation and side elevation of a sphere.
Answer: All three are circles: a sphere looks the same from every direction.
3. A cuboid is 4 cm wide, 3 cm deep and 2 cm tall. What are the dimensions of its plan?
Answer: 4 cm by 3 cm: the plan shows width by depth (the height is lost looking down).
Teacher notes and timings
- Rough timing: Lesson 1 the three views and the everyday solids (sections 1 and 2), Lesson 2 cube-solid counting plus the exit ticket (section 3).
- The isometric figures and every 2D view in this unit are drawn by the PlanElevation engine (components/StandardFigures.tsx) from a single height map per solid, so the solid and its views can never disagree; the same height maps drive the worksheet answer keys.
- Side-elevation convention, stated wherever it matters: this unit's side elevations are taken from the RIGHT of the solid, which places the solid's front edge on the LEFT of the drawn view. Square COUNTS are the same from either side, which is why the worksheets ask for counts.
- Freehand isometric drawing practice (squared/isometric paper) is a valuable classroom follow-on this unit deliberately does not generate worksheets for: a hand drawing has no computed, never-wrong answer key. The 'construct' half of Geometry item 13 is covered here through the build-and-check extension activities instead.
- The cube-solid worksheet generator only produces arrangements in which no cube is hidden (every front stack is shorter than the stack behind it), and each prompt still states that no cubes are hidden, matching exam convention.