Plans and scale
A road map with a scale of 1:250 000. The plan of your future "gaming room" at a scale of 1:40. Both use the same tool: a ratio that relates the drawing to reality. Mastering it lets you work out real distances, areas and even volumes from a simple drawing.
The scale of a map or plan
On a road map of Spain, the given scale is $1:250.000$. If you use a ruler to measure the distance between two towns and measure exactly $8\text{ cm}$, how many…
On a road map of Spain, the given scale is $1:250.000$. If you use a ruler to measure the distance between two towns and measure exactly $8\text{ cm}$, how many kilometres apart are these two towns in reality?
On the floor plan of a house drawn to a scale of $1:100$, the main bedroom is represented as a rectangle measuring $4\text{ cm}$ long and $3\text{ cm}$ wide. Calculate…
On the floor plan of a house drawn to a scale of $1:100$, the main bedroom is represented as a rectangle measuring $4\text{ cm}$ long and $3\text{ cm}$ wide. Calculate the actual dimensions of the room expressed in metres.
From the plan to reality, and back again
You see a linear scale on a hiking map where a segment of $1\text{ cm}$ is equivalent to $500\text{ metres}$ in reality. If the route you want to do measures…
You see a linear scale on a hiking map where a segment of $1\text{ cm}$ is equivalent to $500\text{ metres}$ in reality. If the route you want to do measures $12\text{ cm}$ on the map, how many kilometres will you walk in total?
You have decided to redecorate your room and want to draw a scale plan at $1:20$ to see if a new desk fits. If your room is rectangular and measures $4,2$ metres long by…
You have decided to redecorate your room and want to draw a scale plan at $1:20$ to see if a new desk fits. If your room is rectangular and measures $4,2$ metres long by $3,4$ metres wide, what will the dimensions (length and width) of the rectangle in your drawing be, expressed in centimetres?
Changing scale
Sometimes you need to go from one scale to another: for example, to print a smaller map.
A biologist observes an ant through a microscope. In their technical drawing, the ant has a length of $12$ cm. If the scale of the drawing is an enlargement of $25:1$,…
A biologist observes an ant through a microscope. In their technical drawing, the ant has a length of $12$ cm. If the scale of the drawing is an enlargement of $25:1$, what is the actual length of the ant expressed in millimetres?
A forest monitoring drone flies along a rectangular route over a nature park. At the control centre, technicians view the route on a digital screen where the scale is…
A forest monitoring drone flies along a rectangular route over a nature park. At the control centre, technicians view the route on a digital screen where the scale is $1:50.000$. On this screen, the rectangle measures $12\text{ cm}$ long and $8\text{ cm}$ wide. However, for the printed report, they must reduce the map so that the length of the route is only $4\text{ cm}$ long.
Calculate:
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The actual area of the nature park in square kilometres ($\text{km}^2$).
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The numerical scale that the map in the printed report should have.
Scale and areas
A scale $1:k$ multiplies lengths by $k$, but areas are multiplied by $k^2$, because an area is the product of two lengths.
On a hiking map of the Sierra de Guadarrama, the scale indicated is $1:25.000$. If the distance measured with a ruler between the Puerto de Navacerrada and the Bola del…
On a hiking map of the Sierra de Guadarrama, the scale indicated is $1:25.000$. If the distance measured with a ruler between the Puerto de Navacerrada and the Bola del Mundo is $12,4$ cm on paper, how many actual kilometres will a group of hikers have to travel to go from one point to the other in a straight line?
You are designing your own "Gaming Room" on a scale plan at $1:40$. In the drawing, the room has an area of $75\text{ cm}^2$. You have seen a professional desk that…
You are designing your own "Gaming Room" on a scale plan at $1:40$. In the drawing, the room has an area of $75\text{ cm}^2$. You have seen a professional desk that takes up an actual area of $1,28\text{ m}^2$ and a chair that takes up $0,6\text{ m}^2$.
If you decide to place the desk and two identical chairs, what actual area will remain free in the room? Express your answer in square metres ($\text{m}^2$).