Scale Factors and Maps

Scale Factors and Maps

A scale factor is the ratio by which all lengths in a shape or diagram are multiplied to produce an enlarged or reduced version. Maps and scale diagrams use a stated scale to relate drawn lengths to real distances. Reading and applying scales is a practical application of ratio and proportion.

Scale Notation

A map scale is given as a ratio, e.g. 1 : 50 000. This means every 1 unit on the map represents 50,000 of the same unit in reality. Both sides of the ratio are in the same unit — so 1 cm on the map = 50,000 cm in reality.

  • Map distance → real distance: multiply by the scale factor.
  • Real distance → map distance: divide by the scale factor.
Map (1 : 50 000) 4 cm drawn distance × 50 000 Real distance 200 000 cm = 2 km 4 cm × 50 000 = 200 000 cm = 2 000 m = 2 km

Enlargement Scale Factors

When a shape is enlarged by scale factor \( k \):

  • All lengths are multiplied by \( k \).
  • Area is multiplied by \( k^2 \).
  • Volume is multiplied by \( k^3 \).
  • \( k > 1 \): enlargement. \( 0 < k < 1 \): reduction. \( k < 0 \): enlargement with rotation through 180°.

Finding the scale factor: \( k = \dfrac{\text{image length}}{\text{original length}} \) using any corresponding pair of lengths.

Worked Examples

A map has scale 1 : 25 000. A road measures 8 cm on the map. How long is the road in km?
\[ 8 \times 25{,}000 = 200{,}000 \text{ cm} = 2{,}000 \text{ m} = 2 \text{ km} \]
A park is 3 km long in reality. On a 1 : 50 000 map, how long is it in cm?
\[ 3 \text{ km} = 300{,}000 \text{ cm} \implies 300{,}000 \div 50{,}000 = 6 \text{ cm} \]
A rectangle of length 5 cm is enlarged to length 15 cm. What is the scale factor? What is the new width if the original width is 3 cm?
\[ k = 15 \div 5 = 3 \qquad \text{New width} = 3 \times 3 = 9 \text{ cm} \]
A shape has perimeter 20 cm. After a scale factor of \( \frac{1}{4} \), what is the new perimeter?
\[ 20 \times \frac{1}{4} = 5 \text{ cm} \]
A rectangle has area 12 cm². It is enlarged by scale factor 3. What is the new area? If a similar solid has volume 8 cm³ and is enlarged by scale factor 2, what is the new volume?
\[ \text{New area} = 12 \times 3^2 = 12 \times 9 = 108 \text{ cm}^2 \] \[ \text{New volume} = 8 \times 2^3 = 8 \times 8 = 64 \text{ cm}^3 \]

 Key Takeaways

  • Map scale 1 : n means 1 unit on map = n units in reality. Map → real: multiply by n. Real → map: divide by n.
  • Always work in consistent units — convert both distances to cm (or both to km) before applying the ratio.
  • Scale factor \( k \): lengths × \(k\), areas × \(k^2\), volumes × \(k^3\).
  • Finding \( k \): divide any image length by the corresponding original length.
  • A scale factor between 0 and 1 produces a reduction; greater than 1 produces an enlargement.