Section 12.5: Scale Drawings

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

Examples

• A map has a scale factor of $\frac{1}{10000}$. A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life.
• A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$.
• To reduce a 12-foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$, its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.

Explanation

The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).

Property

To find missing measurements, set up a simple fraction proportion: $\frac{\text{Drawing}}{\text{Actual}} = \frac{\text{Drawing}}{\text{Actual}}$.
Crucial Rule: If your scale does not specify units (like 1:50), you MUST make sure both your drawing and actual measurements are converted to the exact same unit before doing the math.

Examples

• Finding Actual Distance (Unit Form): A map scale is 1 cm = 10 km. Two cities are 4.5 cm apart on the map.
  • Math: 4.5 * 10 = 45. The cities are 45 km apart.
• Finding Actual Distance (Ratio Form): A model boat has a scale of 1:30. The model is 15 cm long. How long is the real boat in meters?
  • Step 1: 15 cm * 30 = 450 cm (Real boat length in cm).
  • Step 2: Convert to meters. 450 cm = 4.5 m.
• Finding Drawing Length: A room is actually 14 feet wide. The blueprint scale is 1 inch = 4 feet.
  • Math: 14 / 4 = 3.5. Draw it 3.5 inches wide on the paper.

Explanation

When solving these, always keep your labels on your numbers! If you write $\frac{1 \text{ in}}{4 \text{ ft}} = \frac{x \text{ in}}{14 \text{ ft}}$, you can clearly see that you need to cross-multiply or divide. If a problem gives you a unitless ratio like 1:300 but asks for the real answer in meters, always calculate the real answer in the tiny unit (like centimeters) first, and do the conversion at the very end.

Property

Scale notations express the relationship between drawing measurements and actual measurements in three common formats: ratio form ($1:100$), unit form ($1 \text{ inch} = 5 \text{ feet}$), and fraction form ($\frac{1}{100}$).

Examples