Plans and Elevations — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers the three flat drawings used to describe a solid object: the plan, the front elevation and the side elevation. By the end of this guide you should be able to say which direction each view is taken from, work out each view for the standard solids, and use the three views together to identify a shape you have not been shown.
You should also be able to draw a solid on isometric paper, understand why internal edges appear as lines on a view, and recognise that different solids can share a view — which is why all three are needed.
The organising idea is that a view is a flattened shadow, taken along one direction. Looking down a cylinder, its curved surface collapses to nothing and only the circular end remains. Looking at the same cylinder from the front, the circle collapses to a straight edge and a rectangle remains. Nothing about the object changes — only the direction you look from — and once you can picture the collapse, every view in this topic follows without needing to memorise a list.
Key terms and definitions
Plan — the view looking straight down from directly above. Sometimes called a bird's-eye view.
Front elevation — the view looking horizontally at the front of the solid.
Side elevation — the view looking horizontally from one side.
Elevation — any horizontal view. The plan is the only one taken from above.
Isometric paper — a triangular grid on which a 3D drawing can be made with lengths kept to scale.
2D representation — any flat drawing standing in for a three-dimensional object.
Core concepts
The three directions
Three views describe a solid because three directions are mutually at right angles.
The plan is taken from above, looking down.
The front elevation is taken from the front, looking horizontally.
The side elevation is taken from the side, also looking horizontally.
Each view shows the outline you would see from that direction, drawn flat, with no perspective and no attempt to show depth.
Why all three are needed
A single view is never enough, because different solids can produce the same one.
A cube, a cylinder lying on its side and a square-based prism can all give a square as one of their views. Only when the other views are added does the solid become identifiable.
This is why exam questions give you two views and ask for the third, or give all three and ask what the solid is.
The standard solids
Working these out by imagining the collapse is more reliable than learning them, but they are worth knowing.
A cube gives a square from every direction.
An upright cylinder gives a circle as its plan, because you are looking down the tube and only the circular end shows. Its front elevation and side elevation are both rectangles, since the circular cross-section collapses to a straight edge when viewed from the side.
This pair is the most commonly tested and the most commonly muddled: the circle is the plan, not the front elevation.
A sphere gives a circle from every direction, which is a property no other standard solid has.
A square-based pyramid standing point-upwards gives a triangle as its front elevation and as its side elevation. Its plan is a square with its diagonals drawn in, because looking down you see the four sloping edges running from the corners to the apex at the centre.
A cone standing point-upwards gives a triangle as both elevations and a circle as its plan.
Lines inside a view
A view is not just an outline. Where an edge of the solid runs along the viewing direction, or where the surface changes direction, a line appears inside the shape.
The diagonals in the plan of a pyramid are exactly this: they are the sloping edges, seen from directly above. Leaving them out is the usual reason a pyramid plan loses its mark.
Similarly, a solid made of stacked cubes shows lines where one block steps back from another.
Drawing the views
Keep the three drawings the same size as each other where they share a dimension. The plan and the front elevation both show the object's width, so those widths must match across the two drawings.
Use a ruler, keep to the squares of the paper if it is squared, and label each view.
Isometric paper
Isometric paper has a triangular grid, which lets a 3D solid be drawn with its lengths kept accurate rather than foreshortened.
Vertical edges are drawn vertically, and the two horizontal directions are drawn along the two sloping grid lines. Lengths along any of these three directions are to scale, which is what makes the drawing measurable rather than merely pictorial.
The grid must be used with the dots or lines running in the intended orientation — drawing on it sideways distorts the solid.
Worked examples
Example 1: An upright cylinder
State the plan, front elevation and side elevation of a cylinder standing on one of its circular ends.
Looking down, the curved surface collapses and the circular end is what remains. The plan is a circle.
Looking from the front, the circular ends collapse to straight edges at the top and bottom, and the curved surface fills the space between them. The front elevation is a rectangle.
Looking from the side, the situation is identical, since the cylinder looks the same from every horizontal direction. The side elevation is also a rectangle.
Example 2: A square-based pyramid
State the three views of a square-based pyramid standing with its apex upwards.
The front elevation is a triangle, formed by the sloping faces seen edge-on.
The side elevation is also a triangle, because the base is square and so the pyramid looks the same from front and side.
The plan is a square with both diagonals drawn. The square is the base, and the diagonals are the four sloping edges seen from directly above, meeting at the apex in the centre.
Example 3: Identifying a solid from its views
A solid has a circular plan, a triangular front elevation and a triangular side elevation. What is it?
A circular plan rules out anything with a flat-sided base, so the solid has a circular cross-section when viewed from above.
Triangular elevations mean it tapers to a point as it rises.
The solid is a cone standing point-upwards. Note that no single one of these three views would have been enough: a circle alone would also fit a cylinder or a sphere.
Common mistakes and how to avoid them
Swapping the cylinder's views. The circle is the plan and the rectangles are the elevations. Picture looking down the tube.
Leaving the diagonals out of a pyramid's plan. Those lines are the sloping edges seen from above, and they are part of the answer.
Confusing plan with front elevation. The plan is the only view taken from above.
Drawing in perspective. Views are flat outlines with no depth and no vanishing points.
Making the three views different sizes. Shared dimensions must match between views.
Giving one view when three were asked for. Read the question — "draw the plan and elevations" means all three.
Assuming one view identifies a solid. Several different solids share any given view.
Exam technique for "Plans and Elevations"
Label each drawing with its name. An unlabelled set of correct views can lose marks because the examiner cannot tell which is which.
Use a ruler and keep to the grid lines if the paper is squared, since these questions carry accuracy marks.
Line the drawings up so that matching dimensions are the same length across views. It makes errors visible to you as well as to the examiner.
Include internal lines where edges are visible from that direction, especially for pyramids and for solids built from stacked cubes.
When identifying a solid from its views, use all three before answering, and check your answer by working the views back out from the solid you have named.
State views as named shapes — "a rectangle", "a square with its diagonals" — rather than describing them loosely.
Quick revision summary
A view is a flattened shadow taken along one direction: what is thin in that direction collapses away.
The plan is from directly above; the front elevation and side elevation are horizontal views from the front and the side.
An upright cylinder has a circular plan and rectangular elevations. A sphere is a circle from every direction. A cube is a square from every direction.
A square-based pyramid has triangular elevations and a plan that is a square with both diagonals drawn — those diagonals are its sloping edges seen from above. A cone has triangular elevations and a circular plan.
Views include internal lines wherever an edge is visible from that direction.
Three views are needed because different solids share any single one.
Isometric paper keeps lengths to scale along its three grid directions, so a 3D drawing on it can be measured.