Scale Drawings & Building Plans
Master scale drawings and building plans for NSW Year 12 Mathematics Standard 2. A scale such as \(1:100\) means every \(1\) unit on the drawing represents \(100\) of the same units in reality, so you can recover real lengths, floor areas and costs from a plan.
You will learn to interpret a scale, convert between drawing and real lengths, write a scale in the form \(1:x\), and find real floor areas and material costs from a floor plan or building plan — a core Standard 2 skill for reading builders’ plans and site plans.
Theory
Scale drawings and building plans show a real object at a fixed ratio, such as \(1:100\). This Year 12 Standard 2 (NSW) guide shows how to interpret a scale, convert between drawing and real lengths, read a floor plan, and find real floor areas and material costs from a plan.
A scale drawing represents a real object at a fixed ratio, so that shapes are kept in proportion but sizes are reduced. A scale such as \(1:100\) means every \(1\) unit measured on the drawing stands for \(100\) of the same units in reality. The number after the colon is the scale factor.
A building plan is a set of scale drawings of a building. A floor plan is the view looking straight down from above, showing the layout and sizes of the rooms; an elevation is the view of one side (front, back or side) showing heights. Both are drawn to a stated scale, so real lengths can be recovered from them.
To find a real length, multiply the plan length by the scale factor; to find a plan length, divide the real length by it. For a real area, convert each side to its real length first, then multiply. This is a core Year 12 Standard 2 (NSW) Ratios and rates skill.
To turn a drawing length into a real length, multiply by the scale factor \(n\):
To go the other way, from a real length to a drawing length, divide by the scale factor (working in the same small unit):
A real area is found from the real side lengths — convert each side first, then multiply:
How to find real measurements from a plan
- Read the scale (for example \(1:100\)) and note the scale factor \(n\).
- Measure the length on the plan in millimetres or centimetres.
- Multiply the plan length by \(n\) to get the real length in that same small unit.
- Convert to sensible units — \(\text{mm}\div 1000\) or \(\text{cm}\div 100\) for metres.
- Area or cost: do this for each side, multiply the real lengths for the area, then multiply the area by any cost-per-square-metre rate.
Multiply the plan length by the scale factor \(50\), then convert to metres.
| \(\text{real}\) | \(=\) | \(50 \times 70\text{ mm}\) |
| \(\) | \(=\) | \(3500\text{ mm}\) |
| \(\) | \(=\) | \(3500 \div 1000\text{ m}\) |
| \(\therefore\ \text{real}\) | \(=\) | \(3.5\text{ m}\) |
Put the real length in centimetres, then divide by the scale factor \(150\).
| \(9\text{ m}\) | \(=\) | \(900\text{ cm}\) |
| \(\text{drawing}\) | \(=\) | \(900 \div 150\text{ cm}\) |
| \(\therefore\ \text{drawing}\) | \(=\) | \(6\text{ cm}\) |
Convert both lengths to the same unit, then simplify so the drawing side is \(1\).
| \(3\text{ m}\) | \(=\) | \(300\text{ cm}\) |
| \(2\text{ cm} : 300\text{ cm}\) | \(=\) | \(1 : 150\) |
| \(\therefore\ x\) | \(=\) | \(150\) |
Convert each side to its real length, find the area, then multiply by the rate.
| \(\text{length}\) | \(=\) | \(100 \times 70\text{ mm} = 7\text{ m}\) |
| \(\text{width}\) | \(=\) | \(100 \times 45\text{ mm} = 4.5\text{ m}\) |
| \(A\) | \(=\) | \(7 \times 4.5 = 31.5\text{ m}^2\) |
| \(\text{cost}\) | \(=\) | \(31.5 \times \$60\) |
| \(\therefore\ \text{cost}\) | \(=\) | \(\$1890\) |
Common pitfalls
Frequently asked questions
What does a scale of 1:100 mean?
It means that 1 unit measured on the drawing represents 100 of the same units in reality. So 1 mm on the plan is 100 mm (10 cm) in real life, and 1 cm on the plan is 100 cm (1 m). The number after the colon, here 100, is the scale factor.
How do you find the real length from a scale drawing?
Measure the length on the drawing, then multiply it by the scale factor. For example at a scale of 1:100 a wall drawn 60 mm long is 60 times 100, which is 6000 mm, and dividing by 1000 gives 6 m in reality.
How do you convert a real measurement to a scale drawing?
Divide the real length by the scale factor, working in the same small unit. For example a room 9 m long on a 1:150 plan is 900 cm divided by 150, which is 6 cm, so you draw it 6 cm long.
How do you work out area from a floor plan?
Do not scale the area directly. Convert each side length from the plan to its real length first, then multiply the real lengths together. For a 1:100 plan measuring 70 mm by 45 mm the real sides are 7 m and 4.5 m, so the real floor area is 7 times 4.5, which is 31.5 square metres.
What is the difference between a floor plan and an elevation?
A floor plan is the view looking straight down from above; it shows the layout and the sizes of the rooms. An elevation is a view of one side of the building, such as the front or side, and shows heights like windows and doors. Both are drawn to a stated scale.
How do you write a scale as a ratio like 1:x?
Put the drawing length and the real length in the same unit, then simplify so the drawing side becomes 1. For example 2 cm representing 3 m becomes 2 cm to 300 cm, which simplifies to 1 to 150, so the scale is 1:150.