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Similarity and Scale

General Mathematics

About these notes

Area Of Study

Geometry and measurement

Unit

Unit 3: General Mathematics Unit 3

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Similarity and Scale

General Mathematics
01 May 2026

Similarity and Scale Diagrams

Similar Figures

Two figures are similar if:
1. All corresponding angles are equal, and
2. All corresponding sides are in the same ratio (proportion).

The ratio is called the scale factor \(k\):

\[k = \frac{\text{image (new) length}}{\text{original length}}\]

Scale Factor and Its Effects

Measurement Effect of scale factor \(k\)
Length Multiply by \(k\)
Area Multiply by \(k^2\)
Volume Multiply by \(k^3\)

Worked Example — Finding a Missing Length

Two similar triangles: Triangle 1 has sides 6 cm, 8 cm, 10 cm. Triangle 2 has a corresponding side of 15 cm (to the 6 cm side).

Scale factor: \(k = \dfrac{15}{6} = 2.5\)

Other sides of Triangle 2: \(8 \times 2.5 = 20\) cm and \(10 \times 2.5 = 25\) cm.

Worked Example — Area Ratio

Two similar rectangles. The smaller has area \(12 \text{ cm}^2\). The scale factor from small to large is \(k = 3\).

\[\text{Larger area} = 12 \times 3^2 = 12 \times 9 = 108 \text{ cm}^2\]

Scale Diagrams and Maps

A scale diagram represents a real object or region at a reduced (or enlarged) size. The scale is expressed as a ratio, e.g., \(1 : 500\) meaning 1 cm on the diagram = 500 cm (5 m) in reality.

\[\text{Real distance} = \text{scale distance} \times \text{scale factor}\]

Worked Example

On a scale diagram with scale \(1:250\), a wall measures 4.8 cm.

\[\text{Real length} = 4.8 \times 250 = 1200 \text{ cm} = 12 \text{ m}\]

Constructing a Scale Diagram

  1. Choose an appropriate scale (e.g., 1 cm : 5 m).
  2. Divide all real dimensions by the scale factor.
  3. Draw accurately using ruler, compass, or CAS geometry.

Confirming Similarity

To confirm two triangles are similar, show one of:
- AA: Two pairs of equal angles (angle-angle)
- SSS: All three pairs of sides in proportion
- SAS: Two pairs of proportional sides and the included angle equal

APPLICATION: Scale diagrams are used in architecture, engineering, cartography, and surveying. VCAA questions may ask you to find a real measurement from a scale diagram or vice versa.

COMMON MISTAKE: Forgetting to square (or cube) the scale factor for area (or volume). Length \(\times k\), area \(\times k^2\), volume \(\times k^3\).

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