Scale Diagrams
Definitions
Discover
When designing a house, an architect doesn’t draw the house at its actual size. That would be far too big to fit on paper! Instead, the architect draws a smaller version of the house where every measurement is reduced by the same amount, called the scale. For house plans, a scale of 1:100 is often used, meaning the house is drawn 100 times smaller than it really is.\includegraphics[width=0.5\textwidth]{\imagepath floorplan2.png} These smaller versions are called scale diagrams.
Definition Scale Diagram
A scale diagram is a method of representing an object at a different proportion to its real-world size using a scale, which is a ratio expressed as 1:scale factor or 1/scale factor.$$\begin{aligned}\dfrac{\textcolor{colorprop}{1}}{\textcolor{colordef}{\text{Scale factor}}}&=\dfrac{\textcolor{colorprop}{\text{Drawn length}}}{\textcolor{colordef}{\text{Actual length}}}\end{aligned}$$
Formulae
Proposition Formulae
$$\begin{aligned}\text{Actual length} &= \text{Drawn length} \times \text{Scale factor}\\\text{Drawn length} &= \text{Actual length} \div \text{Scale factor}\\\text{Scale factor} &= \dfrac{\text{Actual length}}{\text{Drawn length}}\end{aligned}$$
$$\begin{aligned}\dfrac{\textcolor{colorprop}{1}}{\textcolor{colordef}{\text{Scale factor}}}&=\dfrac{\textcolor{colorprop}{\text{Drawn length}}}{\textcolor{colordef}{\text{Actual length}}}\\\textcolor{colorprop}{1} \times \textcolor{colordef}{\text{Actual length}} &= \textcolor{colorprop}{\text{Drawn length}} \times \textcolor{colordef}{\text{Scale factor}} &&\text{(cross multiplication)}\\\textcolor{colordef}{\text{Actual length}} &= \textcolor{colorprop}{\text{Drawn length}} \times \textcolor{colordef}{\text{Scale factor}} &&\text{(simplification)}\end{aligned}$$
Example
Find the width of this house:
The drawn width of the house is \(4 \mathrm{cm}\).$$\begin{aligned}\text{Actual width} &= \text{Drawn width} \times \text{Scale factor}\\&= 4 \mathrm{cm} \times 200 \\&= 800 \mathrm{cm} \\&= 8 \mathrm{m}\end{aligned}$$The actual width of the house is 8 meters.
Example
For the scale \( 1:200 \), find the drawn length corresponding to an actual length of \( 6 \mathrm{m} \).
$$\begin{aligned}\text{Drawn length} &= \dfrac{\text{Actual length}}{\text{Scale factor}}\\&= \dfrac{6 \mathrm{m}}{200}\\&= \dfrac{600 \mathrm{cm}}{200} &&(\text{unit conversion})\\&= 3 \mathrm{cm}\end{aligned}$$So, \(6 \mathrm{m}\) of actual length represents \(3 \mathrm{cm}\) of drawn length.
Example
\(2 \mathrm{cm}\) of drawn length represents \(5 \mathrm{m}\) of actual length.
Find the scale factor.
Find the scale factor.
$$\begin{aligned}\text{Scale factor} &= \dfrac{\text{Actual length}}{\text{Drawn length}}\\&= \dfrac{5 \mathrm{m}}{2 \mathrm{cm}}\\&= \dfrac{500 \mathrm{cm}}{2 \mathrm{cm}} &&(\text{converting to the same units})\\&= 250\end{aligned}$$So, the scale factor is 250.