Chapter 4: Scale Drawings

4.1 Scale Drawings

What is a Scale Drawing?

A scale drawing is a drawing that represents an object such that all dimensions in the drawing are proportional to the actual dimensions of the object.

Scale Ratio Format

Scale is written in the standard ratio form:

$$\text{Scale} = 1 : n$$

where:

$$\text{Scale} = \frac{\text{Measurement of Drawing}}{\text{Measurement of Object}} = \frac{1}{n}$$

Important rule: Both the drawing measurement and object measurement MUST be in the same unit before calculating or stating the ratio.

Interpreting the Value of $n$

• When $n < 1$ (e.g., $1 : \frac{1}{2}$ or $1 : 0.5$): The scale drawing is larger than the actual object.
• When $n = 1$ (i.e., $1 : 1$): The scale drawing is the same size as the actual object.
• When $n > 1$ (e.g., $1 : 2$ or $1 : 500$): The scale drawing is smaller than the actual object.

Key Formulae

1. Finding Scale ($1 : n$):

$$n = \frac{\text{Length of Object}}{\text{Length of Drawing}}$$

2. Finding Drawing Length:

$$\text{Length of Drawing} = \frac{\text{Length of Object}}{n}$$

3. Finding Actual Object Length:

$$\text{Length of Object} = \text{Length of Drawing} \times n$$

Area Relationships in Scale Drawings

If the scale of a drawing is $1 : n$, the relationship between the area of the drawing and the actual area of the object is:

$$\frac{\text{Area of Drawing}}{\text{Area of Actual Object}} = \left(\frac{1}{n}\right)^2 = \frac{1}{n^2}$$ $$\text{Actual Area} = \text{Area of Drawing} \times n^2$$

Steps to Construct a Scale Drawing

1. Determine the scale $1 : n$ and convert all actual lengths to drawing dimensions using $\text{Drawing Length} = \frac{\text{Actual Length}}{n}$.
2. Ensure geometric properties (angles, parallel lines, perpendicular lines) remain unchanged. Angles in scale drawings are always equal to the corresponding angles in the actual object.
3. Draw grid lines or use geometric instruments (ruler, protractor, set squares) to construct the figure accurately.