GCSE · Maths · AQA · Spec 8300 · Foundation

Plans and elevations of 3D shapes

Look down on a box: one rectangle. From the front: another. From the side: a third. Three flat pictures of one solid — and you need all three to rebuild it.

Maths · Plans and elevations

Which way were you looking?

Six drawings. For each one, decide which direction it was seen from — or whether it could never be a plan or elevation of this step at all. Pick a drawing, then pick where it goes.

Still to sort

Plan (0)

Looking straight down from above.

Where the line is: The plan and the front elevation are both 6 cm long. The plan's other measurement is the depth (front to back); the front elevation's is the height.

Front elevation (0)

Looking straight at the front.

Where the line is: From the front you see how long and how tall the step is — never how deep.

Side elevation (0)

Looking straight at the side (here, the right-hand side).

Where the line is: From the side you see how deep and how tall the step is — never how long.

Not a plan or elevation (0)

Could never be seen looking straight at this step from one direction.

6 of 6 still to sort.

Here's our solid for the whole lesson: a step. The bottom block is 6 cm long (left to right), 3 cm deep (front to back) and 2 cm tall. A second block, 2 cm long, 3 cm deep and 2 cm tall, sits on top of its left-hand end — so the left end is 4 cm tall and the rest is 2 cm tall.

Predict, then check

A plan throws away height. So what can a plan on its own really tell you?

A plan shows three squares in a row, each 1 cm by 1 cm. Two students build a solid from 1 cm cubes to match it. Amir builds three cubes in a row. Beth builds the same row, then puts one extra cube on top of the middle cube. Whose solid has this plan?

Watch it done: drawing all three views

Problem

Draw the plan, front elevation and side elevation of this solid on centimetre-squared paper. It's a flat slab 5 cm long, 4 cm deep and 1 cm tall, with a wall 5 cm long, 1 cm deep and 3 cm tall standing along its back edge.

Spot the slip: an elevation drawn to scale

A storage box is 150 cm long, 60 cm deep and 90 cm tall. Its front elevation is drawn to a scale of 1 : 30. How long and how tall is the drawing?

A student's answer — which line goes wrong?

Same step, two kinds of paper

Isometric drawingvsPlan and elevations on square paper

Next you'll use an isometric drawing alongside the plan and elevations. So first: why do they look so different?

Focus

The paper

Isometric drawing

Isometric paper

Plan and elevations on square paper

Normal squared paper

The insight

Plans and elevations don't need special paper — ordinary squared paper does the job.

The grid lines

Isometric drawing

A different grid from ordinary squares

Plan and elevations on square paper

Ordinary squares

The step's 6 cm long edge

Isometric drawing

6 units along the isometric grid

Plan and elevations on square paper

6 squares along the square grid

Your turn: put the drawings together

A step-shaped solid is shown on isometric paper, labelled: 8 cm long, 5 cm tall at the tall end, 2 cm tall at the low end. Its front elevation shows that the tall part is 3 cm long. Its side elevation (from the right) is 4 cm wide. You need to draw the plan. Find its length, its depth, and where the line across it goes.

  1. The plan is the view from above: it shows length and depth, plus a line wherever the top changes height.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One solid, three flat views: from above, from the front and from the side.

What you need to know

  • Plan = the view from above. Front elevation = the view from the front. Side elevation = the view from the side.
  • Each view shows two measurements: plan → length and depth; front elevation → length and height; side elevation → depth and height.
  • Draw a line inside a view wherever you'd see an edge from that direction — usually where the solid changes height.
  • You need all three views to build a solid, and none of them shows what's inside it.
  • Isometric paper and squared paper have different grid lines, so the drawings look different — but every length measures the same.
  • For a scale like 1 : 50, divide real lengths by 50 to get drawn lengths, and multiply drawn lengths by 50 to get real ones.

The big picture

A plan is the view of a solid from above, the front elevation is the view from the front and the side elevation is the view from the side. Each view is flat: it keeps two of the solid's measurements and loses the one pointing at you. You need all three to build the solid, and even then they don't show what's inside. The views can be drawn on ordinary squared paper, drawn to scale (given as a ratio or a scale factor), and combined with an isometric drawing to find missing measurements.

Key points

1Ask 'which way am I looking?' — the measurement pointing straight at you disappears from that view.
2The plan and front elevation share the length, the front and side elevations share the height, and the plan and side elevation share the depth.
3Two different solids can have the same plan, so always check the elevations too.
4A scale can be a ratio (1 : 20) or a scale factor (1/20); they give the same instruction.
5To find a missing length, name the dimension, then read it from a view that shows it.

Worked example

Problem

A solid is made of 1 cm cubes stacked on a table, with no gaps. Its plan is an L of three squares: two side by side at the front, and one behind the left-hand one. Its front elevation is 2 cubes tall on the left and 1 cube tall on the right. Its side elevation is 2 cubes tall at the front end and 1 cube tall at the back end. How many cubes are there, and where?

⚠ Watch out

Putting the wrong pair of measurements on a view — for example drawing the plan with the solid's height instead of its depth. Before you draw, ask which measurement points straight at you from that direction: that's the one the view loses.

🧠

Memory hook

Whatever points straight at your eye vanishes. Look down: height vanishes (plan). Look from the front: depth vanishes (front elevation). Look from the side: length vanishes (side elevation).

✓

Check yourself

A cereal box is 20 cm wide, 7 cm deep and 30 cm tall. What are the measurements of its plan, front elevation and side elevation — and which measurement does each view lose?

Flashcards

(12)
What is the plan of a solid?
The view from directly above. It shows the length and the depth.
What is the front elevation of a solid?
The view from the front. It shows the length and the height.
What is the side elevation of a solid?
The view from the side. It shows the depth and the height.
Why is every plan or elevation flat?
It is seen from one direction, so the measurement pointing straight at you disappears. Two dimensions stay, one is lost.
Why can't you build a solid from its plan alone?
The plan loses the height, so different solids — a single cube and a tower of cubes — can share the same plan. You need the elevations too.
Can plans and elevations show what's inside a solid?
No. They only show the outside; a hidden gap or cube in the middle changes none of them.
When do you draw a line inside a plan or elevation?
Wherever you'd see an edge from that direction — usually where the solid changes height. No edge, no line.
Why does an isometric drawing look different from views on squared paper?
The grid lines on the two papers are different. The measurement of every edge stays the same.
What does a scale of 1 : 40 mean?
1 cm on the drawing stands for 40 cm in real life.
Scale 1 : 40. How do you convert between real and drawn lengths?
Real → drawing: divide by 40. Drawing → real: multiply by 40.
What are the two ways a scale might be given?
As a ratio, like 1 : 40, or as a scale factor, like 1/40. They mean the same thing.
How do you find a missing length using several views?
Decide whether it's a length, depth or height, then read it from a view that shows that dimension.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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