Plans and Elevations
This topic covers how to interpret 2D drawings that represent 3D objects. Understanding plans and elevations is essential for visualising a shape's structure and dimensions from different orthogonal viewpoints.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A plan is the view of an object from directly above, showing its 'footprint' and horizontal dimensions (length and width).
- An elevation is the view from the front or a side, showing vertical dimensions (height) and the object's profile from that direction.
- Multiple views (e.g., a plan, front, and side elevation) are required to fully define a 3D shape, as a single view can be ambiguous.
- These 2D drawings are orthogonal projections, meaning there is no perspective; all lines are drawn to scale without depth distortion.
- Curved surfaces on a 3D object often appear as flat shapes in an elevation. For example, the side elevation of a cylinder is a rectangle.
- A dot in the centre of a circular plan typically represents the apex of a cone viewed from above.
Definitions
- Plan
- A 2D technical drawing showing a 3D object as if viewed from directly above.
- Front Elevation
- A 2D drawing showing a 3D object as if viewed from a designated 'front' side.
- Side Elevation
- A 2D drawing showing a 3D object as if viewed from a designated 'side', typically perpendicular to the front view.
Worked example
The diagrams show the plan, front elevation, and side elevation of a solid object made from a block of wood. The front view is looking from direction X, and the side view is from direction Y. A cylindrical hole is drilled all the way through the block. What is the volume of the resulting solid in cm3? PLAN: A 10cm by 6cm rectangle. In its centre is a circle with diameter 4cm. FRONT ELEVATION (from X): A 10cm by 8cm rectangle. SIDE ELEVATION (from Y): A 6cm by 8cm rectangle.
- 1
First, identify the shapes involved.
The elevations and plan show the outer shape is a rectangular prism (a cuboid) and the inner shape is a cylinder that has been removed.
- 2
Determine the dimensions of the cuboid.
The plan gives length = 10cm and width = 6cm.
The elevations give the height = 8cm.
So, the cuboid's dimensions are 10cm x 6cm x 8cm.
- 3
Calculate the volume of the original cuboid:
Volume = length × width × height = 10 × 6 × 8 = 480 cm3 - 4
Determine the dimensions of the cylindrical hole.
The plan shows its diameter is 4cm, so its radius is 2cm.
The hole passes all the way through, so its height is the height of the block, which is 8cm.
- 5
Calculate the volume of the cylinder:
Volume = pi × r2 × h = pi × (22) × 8 = pi × 4 × 8 = 32pi cm3 - 6
Subtract the volume of the removed cylinder from the volume of the original cuboid to find the final volume:
480 - 32pi cm3.
Answer: 480 - 32pi
Common mistakes
- ×Mistaking which dimension corresponds to which view. Remember: height is shown in the elevations, while length and width are defined by the plan.
- ×Forgetting that curved surfaces appear flat in elevations. The side view of a sphere is a circle; the side view of a cylinder is a rectangle.
- ×Confusing the plan (top-down view) with one of the elevations (side views). Always check the labels or context.
- ×Making assumptions from a single view. A circular plan could be a cylinder, a cone, or another complex shape; you must use all views to be certain.
No-calculator tips
- ✓Before any calculation, try to sketch a quick 3D impression of the object. This helps confirm your interpretation of the 2D views.
- ✓Mentally align the views. The width of the front elevation must match the width on the plan. The height of the front and side elevations must be the same.
- ✓When volume or area calculations involve circles, always work with pi as a symbol. Do not substitute a numerical value like 3.14, as answers are typically left in terms of pi.