Plans and scale
A road map of the Dominican Republic with a scale of $1:250{,}000$. The plan of your future bedroom at a scale of $1:40$. Both use the same idea: a ratio that relates the drawing to reality. If you master it, you can calculate real distances, areas, and even volumes from a simple drawing. How many kilometers are there really between two towns that are 8 cm apart on the map? You'll solve it here.
The scale of a map or plan
That's how you answer the opening question: $8 \times 250{,}000 = 2{,}000{,}000$ cm $= 20$ km.
On a road map of the Dominican Republic, the given scale is $1:250,000$. If you use a ruler to measure the distance between two towns and measure exactly $8\text{ cm}$,…
On a road map of the Dominican Republic, 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 kilometers 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 meters.
A model car collector has a scale model of $1:24$. If the length of the miniature is $18\text{ cm}$, how long will the actual car be in meters?
A model car collector has a scale model of $1:24$. If the length of the miniature is $18\text{ cm}$, how long will the actual car be in meters?
From the plan to reality, and back
Graphic scale
Some maps have a bar: for example, $1\text{ cm}$ on the map is equivalent to $500\text{ m}$. Then you use a proportion: if $1\text{ cm}$ is $500\text{ m}$, $12\text{ cm}$ is $12 \times 500 = 6{,}000\text{ m} = 6\text{ km}$. If you need the numerical scale, convert both measurements to the same unit: $1\text{ cm} : 500\text{ m} = 1 : 50{,}000$.
Enlargement
When something is enlarged (microscope, photos of insects), the scale is written $n:1$. A scale of $5:1$ means that the drawing is 5 times larger than the real object: the real measurement is the drawing's measurement divided by $5$.
You see a linear scale on a hiking map where a segment of $1\text{ cm}$ is equivalent to $500\text{ meters}$ 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 $500\text{ meters}$ in reality. If the route you want to do measures $12\text{ cm}$ on the map, how many kilometers 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 $4.2$ meters 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 $4.2$ meters long by $3.4$ meters wide, what will the dimensions (length and width) of the rectangle in your drawing be, expressed in centimeters?
A road map has a linear scale in which a segment of $2$ cm represents an actual distance of $15$ km. If the distance between two cities on that map is $9.4$ cm, what is…
A road map has a linear scale in which a segment of $2$ cm represents an actual distance of $15$ km. If the distance between two cities on that map is $9.4$ cm, what is the actual distance in kilometers between both cities?
Finding the scale
If you know the measurement on the drawing and the real measurement, the scale is their ratio, with both in the same unit.
A group of archaeologists has drawn an ancient vase on paper. In the drawing, the vase is $15\text{ cm}$ high. If we know that the actual vase has a height of…
A group of archaeologists has drawn an ancient vase on paper. In the drawing, the vase is $15\text{ cm}$ high. If we know that the actual vase has a height of $75\text{ cm}$, what scale have the archaeologists used?
In a model of a famous communications tower, the height of the replica is $85$ cm. If we know that the actual height of the tower is $255$ meters, determine the…
In a model of a famous communications tower, the height of the replica is $85$ cm. If we know that the actual height of the tower is $255$ meters, determine the simplified numerical scale to which the model was built.
An old map has lost its numerical scale indication. However, it shows a linear scale in which a line segment measuring $3.5\text{ cm}$ on paper represents an actual…
An old map has lost its numerical scale indication. However, it shows a linear scale in which a line segment measuring $3.5\text{ cm}$ on paper represents an actual distance of $17.5\text{ kilometers}$.
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Determine the numerical scale of the map.
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If two towns are separated on the map by $12\text{ cm}$, how many hours will it take a cyclist to go from one to the other at a constant speed of $24\text{ km/h}$?
Changing scale
Sometimes you have to go from one scale to another: for example, to print a smaller map or when you zoom in on your cell phone.
The same thing happens with a zoom: the real distance between two stops doesn't change, only its measurement on the screen does.
You have two different maps of the same area. On Map A, the distance between two villages is $6$ cm and the scale is $1:50,000$. On Map B, the distance between the same…
You have two different maps of the same area. On Map A, the distance between two villages is $6$ cm and the scale is $1:50,000$. On Map B, the distance between the same villages is $15$ cm. Calculate the numerical scale of Map B.
A forest monitoring drone flies along a rectangular route over a nature park. At the control center, 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 center, 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 kilometers ($\text{km}^2$).
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The numerical scale that the map in the printed report should have.
A photocopy has been made of an original plan that was at a scale of $1:200$. Due to a setting error, the photocopy has come out reduced to $75\%$ of its original size.…
A photocopy has been made of an original plan that was at a scale of $1:200$. Due to a setting error, the photocopy has come out reduced to $75\%$ of its original size.
On the photocopy, the living room of the house measures $6\text{ cm}$ in length.
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What is the actual scale of the photocopy compared to reality?
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What is the actual length of the living room in meters?
Scale and areas
The scale $1:k$ multiplies lengths by $k$, but areas get multiplied by $k^2$, because an area is the product of two lengths.
Another way, sometimes more convenient: convert each side to its real measurement and multiply. A field drawn as $25\text{ cm} \times 15\text{ cm}$ at a scale of $1:400$ measures $100\text{ m} \times 60\text{ m}$, that is, $6{,}000\text{ m}^2$.
The plan of a local soccer field is drawn to a scale of $1:400$. On the plan, the area occupied by the grass is a rectangle $25$ cm long by $15$ cm wide. Calculate the…
The plan of a local soccer field is drawn to a scale of $1:400$. On the plan, the area occupied by the grass is a rectangle $25$ cm long by $15$ cm wide. Calculate the actual area of the soccer field in square meters.
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 meters ($\text{m}^2$).
The plan of a municipal swimming pool is drawn to such a scale that one of its walls, which measures $25\text{ meters}$ in reality, is represented by a line segment of…
The plan of a municipal swimming pool is drawn to such a scale that one of its walls, which measures $25\text{ meters}$ in reality, is represented by a line segment of $12.5\text{ cm}$. On the plan, the bottom of the pool has an area of $120\text{ cm}^2$.
They want to cover the actual bottom with square tiles measuring $20\text{ cm}$ on each side. How many tiles will need to be bought?
Scale and volumes
An architect has built a scale model of a new auditorium at a scale of $1:250$. The total volume of the model is $0.64\text{ dm}^3$. Given that the density of the actual…
An architect has built a scale model of a new auditorium at a scale of $1:250$. The total volume of the model is $0.64\text{ dm}^3$.
Given that the density of the actual materials is different, but the volume remains proportional to the cube of the scale factor, calculate the actual volume of the auditorium in cubic meters ($\text{m}^3$).
(Note: Remember that if the length scale factor is $k$, the volume scale factor is $k^3$).
Problems with scale
Many problems combine scale with speed, time, or screens. Always follow the same path: measure on the drawing, convert to real distance, and then use whatever the problem asks for.
You want to make a drawing in your notebook of your new graphics tablet. The tablet has an actual length of $30\text{ cm}$. If you decide to use a reduction scale of…
You want to make a drawing in your notebook of your new graphics tablet. The tablet has an actual length of $30\text{ cm}$. If you decide to use a reduction scale of $1:3$, what will the length of the drawing in your notebook be?
A biologist photographs a small queen ant using an enlargement scale of $5:1$. If in the photograph the ant measures $45\text{ mm}$ in length, what is its actual size in…
A biologist photographs a small queen ant using an enlargement scale of $5:1$. If in the photograph the ant measures $45\text{ mm}$ in length, what is its actual size in millimeters?
In a sports center brochure, a basketball court has been drawn to a scale of $1:200$. If the actual court is $28\text{ meters}$ long, how many centimeters long will the…
In a sports center brochure, a basketball court has been drawn to a scale of $1:200$. If the actual court is $28\text{ meters}$ long, how many centimeters long will the court be in the drawing in the brochure?
A young architect is designing a new skate park for her city. On the technical plan, the main jump track has a length of $15$ cm. If the park is going to be built to a…
A young architect is designing a new skate park for her city. On the technical plan, the main jump track has a length of $15$ cm. If the park is going to be built to a scale of $1:150$, calculate the actual length of the track in meters.
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 millimeters?
On a road map of Spain with a scale of $1:2,500,000$, the distance measured with a ruler between Madrid and Valencia is $14.2\text{ cm}$. A delivery lorry leaves Madrid…
On a road map of Spain with a scale of $1:2,500,000$, the distance measured with a ruler between Madrid and Valencia is $14.2\text{ cm}$. A delivery lorry leaves Madrid at $08:00\text{ h}$ and travels towards Valencia at an average speed of $90\text{ km/h}$.
What time is it expected to arrive at its destination if it stops for half an hour halfway through to rest?
A map application on a smartphone shows an area of the city. When the zoom is at level A, the scale is $1:15,000$ and the distance between two metro stations is…
A map application on a smartphone shows an area of the city. When the zoom is at level A, the scale is $1:15,000$ and the distance between two metro stations is $6\text{ cm}$. If the user zooms out until the scale is $1:50,000$ (level B):
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What will the distance in centimeters between those same stations be on the screen?
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If the phone screen has an area of $80\text{ cm}^2$, how many actual square kilometers of urban surface are being displayed at level B?