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GCSE Maths · Geometry & measures
Plans and elevations GCSE Questions and Worked Answers
A plan is a view from directly above; an elevation is a view straight from a specified side. Keep the dimensions visible from that direction and hide depth. For solid cube stacks on a level base, each elevation shows the tallest stack along each viewing line.
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Start with the meaning
What you need to know about plans and elevations
Place a cereal box on a table. Looking straight down, you see its top rectangle: this is the plan. Looking straight at the front, you see its front rectangle: this is the front elevation. Looking from the right gives the right elevation. These are flat views, so they do not show perspective or sloping edges receding into the distance.
See the idea first
Choose the viewing direction before drawing
For stacks of identical cubes, a plan shows the occupied floor squares. An elevation shows width and height but hides how far back each cube is. In this example all stacks are solid, with no gaps. From the front, a short stack can be hidden behind a taller stack; use the maximum height in each column, not the sum.
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Draw a plan
What the problem asks: A solid has stacks with heights 2, 0 and 3 in one row. What is its plan?
How to solve it: Draw a square for the first and third positions, leaving the middle position empty. Stack heights do not change the footprint.
Draw an elevation
What the problem asks: Front row heights are 1, 3, 1; back row heights are 2, 1, 0. What are the front-elevation heights?
How to solve it: Look through each column from the front. The maximum heights are max(1,2) = 2, max(3,1) = 3, max(1,0) = 1. Draw columns 2, 3, 1 cubes high.
Count cubes or test uniqueness
What the problem asks: Can the ordinary plan alone tell you how many cubes there are?
How to solve it: No. It identifies occupied floor squares, not stack heights. A numbered height map gives more information; if stacks are solid, add the heights to count the cubes.
A reliable routine
Project solid stacks into a specified view
Use the maximum-height method for stacks rising from the same base without overhangs or gaps. It works because looking straight along a row hides depth and reveals the full silhouette of the tallest stack. Other solids need their actual visible outlines.
- Mark front, back, left and right from the question's arrow.
- For the plan, mark occupied floor positions.
- For a front elevation, keep the greatest height in each left-to-right column; for a right elevation, find the maximum within each row, then list the rows from front to back.
- Draw square edges without perspective and check the dimensions against the chosen view.
Check: Unless the diagram explicitly gives stack heights and no gaps, do not invent hidden cubes. Different 3D shapes can share the same plan and elevations.
Fully worked
Plans and elevations GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Views of a cuboid
Question
A cuboid is 6 cm wide, 4 cm deep and 3 cm high. State the dimensions of its plan, front elevation and right elevation.
Plan keeps width and depth: 6 × 4 cm. Front keeps width and height: 6 × 3 cm. Right keeps depth and height: 4 × 3 cm. Each view hides the dimension pointing towards the viewer.
Example 2
Footprint from heights
Question
A height map has back row [2, 0, 1] and front row [1, 3, 1], left to right. Describe its ordinary plan.
All three front squares are occupied. In the back row only left and right squares are occupied; the middle square is empty.
Do not draw three squares on top of each other for height 3.
Example 3
Front elevation
Question
A different height map has solid stacks with back row [2, 1, 0] and front row [1, 3, 1]. Give front-elevation heights left to right.
Looking from the front compares front and back in each column.
Draw columns of heights [2, 3, 1]. Do not add front and back heights.
Example 4
Right elevation
Question
Use back row [2, 1, 0] and front row [1, 3, 1]. Give right-elevation heights in front-to-back order.
Looking from the right compares left, middle and right stacks within a row. Front-row maximum = 3; back-row maximum = 2. The right elevation has heights [3, 2] in the requested front-to-back order. Standing on the right and looking straight at the solid, its front edge appears on your left.
Example 5
Count solid stacks
Question
A height map has back row [2, 1, 0] and front row [1, 3, 1]. All stacks are solid from a common base. How many cubes are present?
Add the number of cubes in each stack:
There are 8 cubes. The no-gaps condition makes adding heights valid.
Example 6
Same views, different solids
Question
A fully occupied 2 × 2 footprint has stack heights back [2, 1], front [1, 2]. Another has back [2, 2], front [2, 2]. Explain why their plans and front/right silhouettes match but their cube counts differ.
Both plans contain four squares. Each front-view column has maximum 2, and each right-view row has maximum 2, in both solids. But the counts are
and
Matching silhouettes do not uniquely determine hidden stack heights.
10 original questions · total 18 marks
Plans and elevations GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 23 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Name a view
What is the mathematical name for a view directly from above?
Show worked answer
The plan. It preserves width and depth, while hiding height.
Cuboid front
A cuboid is 8 cm wide, 5 cm deep and 2 cm high. Give its front-elevation dimensions.
Show worked answer
8 cm wide by 2 cm high. Depth is hidden.
Cuboid right
For the same stated dimensions 8 cm wide, 5 cm deep, 2 cm high, give the right-elevation dimensions.
Show worked answer
5 cm deep by 2 cm high, shown as a 5 × 2 cm rectangle.
Occupied squares
A height map has back [0, 2, 1] and front [1, 1, 0]. How many squares are in its plan?
Show worked answer
Count non-zero positions: two in each row.
Four floor squares are occupied.
Front maximum
Solid stacks have back [3, 1] and front [1, 2]. Give front-elevation heights left to right.
Show worked answer
The heights are [3, 2].
Right maximum
Solid stacks have back [3, 1] and front [1, 2]. Give right-elevation heights in front-to-back order.
Show worked answer
Front maximum is 2; back maximum is 3. The heights are [2, 3] in that order.
Cube count
Solid stacks have back [3, 1] and front [1, 2], with no gaps. Count the cubes.
Show worked answer
There are 7 cubes.
A cylinder
A cylinder of radius 3 cm and height 8 cm stands upright. Describe its plan and front elevation.
Show worked answer
The plan is a circle of radius 3 cm (diameter 6 cm). The front elevation is a rectangle 6 cm wide and 8 cm high: its width is the cylinder's diameter.
Insufficient information
A plan shows three occupied squares in one row, but gives no heights. Can you determine the number of cubes?
Show worked answer
No. Assuming stacks of cubes, there are at least three cubes, but stacks could be taller. The ordinary plan supplies no heights.
A missing height
Two solid stacks lie one behind the other. The front stack has height 2. The front elevation has height 4. What is the back stack's height?
Show worked answer
The elevation is the maximum of the two heights. Since the front is only 2, the back must have height 4. It cannot be above 4, which would make the elevation taller. It cannot be below 4 either: then neither stack would reach the required height of 4.
Examiner-style feedback
Common plans and elevations mistakes
For solid stacks, an elevation uses the maximum visible height along a sightline, not the total.
Read the viewing arrow. State your left-to-right or front-to-back order.
Plans and elevations are flat projections, not 3D sketches.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Plan: above.
- Elevation: specified side.
- Keep visible dimensions; hide depth.
- State assumptions before counting hidden cubes.
Quick answers
Plans and elevations FAQ
Should a plan contain height numbers?
Not normally. A height map adds numbers as extra information; it is not the same as an ordinary plan.
Can I reconstruct a unique solid from three views?
Not always. Hidden parts can differ while all three visible projections remain the same.
Content standards
Curriculum and rights review
G13 plans and elevations across tiers. Cube-stack tasks state no gaps and a common base; height-map counts are a teaching aid, not a required feature of a standard plan. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references