Scale Drawing, Plans, and Elevations (HSC SSCE Mathematics Standard): Revision Notes

Scale Drawing, Plans, and Elevations

Introduction to scale drawings

Scale drawings are essential tools that allow us to represent real objects at a smaller, more manageable size while maintaining accurate proportions. These drawings are used extensively in many fields including architecture, engineering, and cartography. The key principle behind scale drawings is that all measurements are reduced or enlarged by the same factor, which means the drawing maintains the same proportions as the actual object.

The scale factor is the ratio that compares the size of the drawing to the actual size of the object. For example, maps are scale drawings that represent large geographical areas on paper. A map of Australia might be only a few centimetres across, but it accurately represents a continent that spans thousands of kilometres.

The measurements on a scale drawing have been systematically reduced to make the representation convenient to use and transport. Understanding how to work with scale drawings means being able to convert between the drawing measurements and actual real-world dimensions.

Understanding scale and scale notation

The scale of a drawing establishes the mathematical relationship between the drawing and reality. This relationship is expressed as a ratio in the following formula:

$\text{Scale of a drawing} = \text{Drawing length} : \text{Actual length}$

There are two standard ways to express scale, and it is important to understand both methods:

Method 1: Using units

This method explicitly states the units for both the drawing and the actual object. For example:

• $1 \text{ cm to } 1 \text{ m}$ (or written as $1 \text{ cm} = 1 \text{ m}$)
• This tells us that every $1$ centimetre on the drawing represents $1$ metre in reality

Method 2: Using ratios without units

This method expresses the scale as a pure ratio. For example:

• $1:100$
• This tells us that the actual distance is $100$ times the length shown on the scale drawing
• Both measurements must be in the same units when using this notation

Converting between drawing and actual measurements

To solve problems involving scale drawings, you need to convert between drawing measurements and actual measurements. The method depends on what information you have and what you need to find.

Finding actual length from drawing length:

• Multiply the drawing measurement by the scale factor
• Remember to convert units if necessary

Finding drawing length from actual length:

• Divide the actual measurement by the scale factor
• Remember to convert units if necessary

Plans and elevations

When representing three-dimensional objects in two dimensions, we use different views to show the complete structure. Plans and elevations are standard views used in technical drawing and architecture to communicate spatial information clearly.

Plan view is the view of an object when looking directly down from above. Imagine you are hovering over an object looking straight down at it. This top-down perspective is particularly useful for showing the layout and arrangement of spaces. In a house plan, you can see the arrangement of rooms, the positions of doors and windows, and the flow between spaces.

Elevation view is the view of an object when looking at it from one side. The most common elevation views are:

• Front elevation: the view when looking at the object from the front
• Side elevation: the view when looking at the object from the side

Unlike plans, which are horizontal sections, elevations are vertical sections that show the height and profile of objects. In house elevations, you can see the roof shape, window positions at different heights, and the overall external appearance of the building.

Building plans

Building plans are one of the most common and practical applications of scale drawings. Architects and builders use floor plans drawn to scale to communicate design ideas and construction specifications. These plans allow large buildings to be represented on manageable sheets of paper while maintaining accurate proportions.

Building plans typically use scale factors such as $1:150$ or $1:100$. This means that every $1$ centimetre on the plan represents $150$ centimetres (or $1.5$ metres) in the actual building. By measuring distances on the plan and applying the scale factor, you can determine the actual dimensions of rooms, walls, and other features.

A floor plan is a horizontal section cut through a building at a certain height, showing:

• The arrangement and dimensions of rooms
• Wall positions and thicknesses
• Locations of doors and their opening directions
• Window positions
• Built-in fixtures and fittings
• Appliances and furniture (sometimes)

Common floor plan symbols

Understanding floor plan symbols is essential for reading building plans. These standardised symbols convey important information about the building's features: