Scale drawings Knowledge-Based Study

Study notes

1. What is a Scale Drawing?

   A scale drawing is a proportional two-dimensional representation of an object that has been enlarged or reduced by a specific factor. Architects, cartographers, and engineers use scale drawings such as floor plans, blueprints, and maps to represent large structures or geographic regions on manageable sheets of paper. Similarly, biologists use scale drawings to magnify microscopic structures.

   The Concept of Scale

   The relationship between the dimensions in the drawing and the corresponding actual dimensions is defined as the scale. A scale is typically written as a ratio:

   $$\text{Scale} = \frac{\text{Drawing Dimension}}{\text{Actual Dimension}}$$

   For example, a scale of 1 cm : 5 m indicates that every 1 centimeter measured on the drawing represents an actual distance of 5 meters in the real world.

   • Reduction: The drawing is smaller than the actual object (e.g., maps, house blueprints).
   • Enlargement: The drawing is larger than the actual object (e.g., a diagram of a cell or a microchip).

   Calculating Real-World Dimensions

   When given a scale drawing and a scale, you can find the actual measurements by setting up and solving a proportion.

   Step-by-Step Method

   1. Identify the given scale ratio: $\frac{\text{drawing length}}{\text{actual length}}$.
   2. Write a proportion equating the scale to the measured drawing distance over the unknown actual distance: $$\frac{\text{Scale Drawing Unit}}{\text{Scale Actual Unit}} = \frac{\text{Measured Drawing Length}}{\text{Actual Length (}x\text{)}}$$
   3. Cross-multiply and solve for the unknown variable $x$.
   4. Ensure the final answer includes the appropriate real-world units.

   Example

   A blueprint of a classroom has a scale of 1 cm : 3 m. On the blueprint, the length of the classroom is 4.5 cm. What is the actual length of the classroom?

   Set up the proportion:

   $$\frac{1\text{ cm}}{3\text{ m}} = \frac{4.5\text{ cm}}{x\text{ m}}$$

   Cross-multiply to solve:

   $$1 \times x = 3 \times 4.5$$

   $$x = 13.5\text{ m}$$

   The actual length of the classroom is 13.5 meters.

   Determining Dimensions for a Scale Drawing

   To produce a scale drawing or find a dimension on a blueprint, reverse the process: divide the actual measurement by the scale factor or solve using a proportion.

   Example with Unit Conversion

   An architect is drawing a floor plan with a scale of 1 cm : 2 m. A patio has an actual length of 7.6 meters. How long should the patio be on the drawing?

   Set up the proportion:

   $$\frac{1\text{ cm}}{2\text{ m}} = \frac{d\text{ cm}}{7.6\text{ m}}$$

   Cross-multiply:

   $$2 \times d = 1 \times 7.6$$

   $$d = \frac{7.6}{2} = 3.8\text{ cm}$$

   The patio should be drawn with a length of 3.8 cm.

   Unitless Scale Factors

   When both quantities in a scale share the same unit, the scale can be simplified into a unitless ratio or scale factor. For instance, if a model car has a scale of 1 inch to 2 feet, convert feet to inches:

   $$2\text{ feet} = 24\text{ inches}$$

   The scale ratio without units is written as 1 : 24, meaning any dimension on the real vehicle is 24 times as large as the corresponding dimension on the model.

   How Scale Affects Area

   A common misconception is assuming that the area of a scaled figure changes by the same factor as the linear dimensions. When every linear dimension of a two-dimensional figure is scaled by a factor $k$, the area of the figure changes by a factor of $k^2$.

   Mathematical Relationship

   If the linear scale is:

   $$\frac{\text{Drawing Length}}{\text{Actual Length}} = k$$

   Then the ratio of their areas is:

   $$\frac{\text{Drawing Area}}{\text{Actual Area}} = k^2$$

   Example

   A rectangular garden on a design plan measures 2 cm by 5 cm. The map scale is 1 cm : 4 m. Find the actual area of the garden.

   • Method 1 (Convert dimensions first):
     Actual width $= 2 \times 4 = 8\text{ m}$
     Actual length $= 5 \times 4 = 20\text{ m}$
     Actual Area $= 8\text{ m} \times 20\text{ m} = 160\text{ m}^2$
   • Method 2 (Using area scale factor):
     Drawing area $= 2\text{ cm} \times 5\text{ cm} = 10\text{ cm}^2$
     Since $1\text{ cm} = 4\text{ m}$, then $1\text{ cm}^2 = 4^2\text{ m}^2 = 16\text{ m}^2$.
     Actual Area $= 10 \times 16 = 160\text{ m}^2$

   Both methods demonstrate that area scales with the square of the linear scale factor.