Scale drawings | Year 11 | Ascent Library

01 / LEARN

Understand the idea.

Understand the idea

A scale of $1 : n$ compares a drawing length with its real length in the same unit. For example, at $1 : 100$ , $1 cm$ on paper represents $100 cm = 1 m$ in reality. The ratio alone does not change centimetres into metres.

The scale factor applies to every length. Areas therefore change by the square of that factor, and volumes by its cube. Converting both side lengths of a rectangle before finding its area is a reliable way to avoid using a linear factor for area.

Choose and apply a method

To recover a real length, multiply the drawing length by the scale denominator. To create a drawing length, divide the real length by that denominator. Convert units before dividing if the drawing must be expressed in a different unit.

Use the measurements stated on a diagram rather than measuring a resized image on screen. Printing or resizing can change its physical scale. If a real paper plan is needed, verify a labelled reference length after printing.

\begin{array}{c} real length = n \times drawing length \\ (same units) \end{array}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read a floor plan

A room is 4.5 cm by 3 cm on a 1:200 plan. Find its actual dimensions and area.

Worked solution

1
Translate the scale into convenient units.
$1 cm on plan 200 cm = 2 m in reality$
The 1:200 ratio uses matching units before conversion to metres.
2
Convert both room dimensions.
$\begin{aligned} l & = 4.5 \times 2 = 9 m \\ w & = 3 \times 2 = 6 m \end{aligned}$
Every plan length is multiplied by the same linear scale factor.
3
Calculate the real floor area.
$A = 9 \times 6 = 54 m^{2}$
Multiplying the real dimensions automatically applies the scale factor in both directions.

Answer

9 m by 6 m; 54 m $2$ .

EXAMPLE 02

Draw a length

A 12 m path is shown at 1:150. Find its drawing length.

Worked solution

1
Put the real length in centimetres.
$12 m = 1200 cm$
The drawing length will be in centimetres, so both parts of the scale comparison need that unit.
2
Divide by the scale denominator.
$L_{drawing} = \frac{1200}{150} = 8 cm$
The real length is 150 times the drawing length.
3
Check by scaling back.
$8 \times 150 = 1200 cm = 12 m$
The proposed drawing length reproduces the stated real length.

Answer

8 cm.

03 / YOUR TURN

Try it yourself.

Check your understanding

At 1:50, what real length does 6 cm represent?

Show answer & reasoning

3 m.

Multiply the plan length by 50 using centimetres: $6 \times 50 = 300 cm$ . Since $100 cm = 1 m$ , the real length is $3 m$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

At 1:100, a plan wall is 7.2 cm long. Find its real length.
2 marks
A 9 m fence is drawn at 1:300. Find its drawing length.
2 marks
A 1:250 plan shows a rectangle 4 cm by 2 cm. Find real area.
2 marks
Similar solids have corresponding lengths in ratio 2:3. Find their volume ratio.
2 marks
A model has area 24 cm $2$ . An enlargement triples every length. Find new area.
2 marks
A 1:10,000 map is reduced to half its original linear size. Find the new scale.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

At 1:100, a plan wall is 7.2 cm long. Find its real length.
2 marks
$7.2 \times 100 = 720$ cm $= 7.2$ m.
7.2 m.
A 9 m fence is drawn at 1:300. Find its drawing length.
2 marks
$\frac{900}{300} = 3$ cm.
3 cm.
A 1:250 plan shows a rectangle 4 cm by 2 cm. Find real area.
2 marks
1 cm represents 2.5 m.
The real rectangle is 10 m by 5 m, so its area is 50 m $2$ .
50 m $2$ .
Similar solids have corresponding lengths in ratio 2:3. Find their volume ratio.
2 marks
Cube the linear factors: $2^{3} : 3^{3} = 8 : 27$ .
8:27.
A model has area 24 cm $2$ . An enlargement triples every length. Find new area.
2 marks
The area factor is $3^{2} = 9$ ; $24 \times 9 = 216$ .
216 cm $2$ .
A 1:10,000 map is reduced to half its original linear size. Find the new scale.
2 marks
The same real length now uses half the paper length, so the denominator doubles.
1:20,000.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics measurement skills and applications; original explanatory lessons and papers.

Official curriculum source ↗

In this lesson

Understand the idea.

Understand the idea

Choose and apply a method

See the method in action.

Read a floor plan

Translate the scale into convenient units.

Convert both room dimensions.

Calculate the real floor area.

Draw a length

Put the real length in centimetres.

Divide by the scale denominator.

Check by scaling back.

Try it yourself.

Check your understanding

Keep practising