Scale Drawings and Maps
Emma's family is planning a road trip across Texas. Her road map says the scale is $1:250{,}000$, and she measures $5$ in. between two towns with a ruler. How far is the drive? Maps, floor plans, and model cars all use the same tool: a ratio that links the drawing to reality. Once you master it, you can find real distances, areas, and even volumes from a simple drawing.
What a scale means
The scale works with any unit: $1:50$ means $1$ inch is $50$ inches, or $1$ cm is $50$ cm. The number $k$ in $1:k$ is called the scale factor.
| Type | Scale | What it means |
|---|---|---|
| Reduction | $1:k$ with $k>1$ | The drawing is smaller than the real thing (maps, floor plans, model cars) |
| Actual size | $1:1$ | Same size |
| Enlargement | $n:1$ with $n>1$ | The drawing is bigger than the real thing (an ant under a microscope) |
A group of archaeologists has drawn an ancient vase on a piece of paper. In the drawing, the vase measures $15\text{ cm}$ high. If we know that the real vase has a…
A group of archaeologists has drawn an ancient vase on a piece of paper. In the drawing, the vase measures $15\text{ cm}$ high. If we know that the real vase has a height of $75\text{ cm}$, what scale have the archaeologists used?
You want to make a drawing in your notebook of your new graphics tablet. The tablet has a real length of $30\text{ cm}$. If you decide to use a reduction scale of $1:3$,…
You want to make a drawing in your notebook of your new graphics tablet. The tablet has a real length of $30\text{ cm}$. If you decide to use a reduction scale of $1:3$, how long will the drawing in your notebook measure?
Working with units
A scale only gives you a number. To get a real distance in miles or feet, you need to convert units. Keep this table handy.
| Customary | Metric |
|---|---|
| $1$ ft $= 12$ in. | $1$ in. $= 2.54$ cm |
| $1$ yd $= 3$ ft $= 36$ in. | $1$ m $= 100$ cm |
| $1$ mi $= 5{,}280$ ft $= 63{,}360$ in. | $1$ km $= 1{,}000$ m $= 100{,}000$ cm |
| $1$ mi $\approx 1{,}609$ m $\approx 1.609$ km |
On a road map of the US, the indicated scale is $1:250.000$. If you use a ruler to measure the distance between two towns and mark exactly $8\text{ in}$, how many miles…
On a road map of the US, the indicated scale is $1:250.000$. If you use a ruler to measure the distance between two towns and mark exactly $8\text{ in}$, how many miles separate these two towns in reality?
On the plan of a house drawn to a scale of $1:100$, the master bedroom is represented as a rectangle measuring $4\text{ in}$ long and $3\text{ in}$ wide. Calculate the…
On the plan of a house drawn to a scale of $1:100$, the master bedroom is represented as a rectangle measuring $4\text{ in}$ long and $3\text{ in}$ wide. Calculate the real dimensions of the room expressed in feet.
In a community center brochure, they have drawn a basketball court at a scale of $1:200$. If the real court measures $28\text{ meters}$ long, how many inches will the…
In a community center brochure, they have drawn a basketball court at a scale of $1:200$. If the real court measures $28\text{ meters}$ long, how many inches will the length of the court measure in the brochure's drawing?
From drawing to real, and back
A miniature car collector has a model at a scale of $1:24$. If the length of the miniature is $18\text{ in}$, how long will the real car measure in feet?
A miniature car collector has a model at a scale of $1:24$. If the length of the miniature is $18\text{ in}$, how long will the real car measure in feet?
A young architect is designing a new skate park* for her city. On the technical plan, the main jump track has a length of $15$ in. If the park is going to be built at 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 $15$ in. If the park is going to be built at a scale of $1:150$, calculate the real length of the track in feet.
You have decided to change the decoration of your room and you want to draw a plan at a scale of $1:20$ to see if a new desk fits. If your room is rectangular and…
You have decided to change the decoration of your room and you want to draw a plan at a scale of $1:20$ to see if a new desk fits. If your room is rectangular and measures $4,2$ feet long by $3,4$ feet wide, what will the dimensions (length and width) of the rectangle in your drawing be, expressed in inches?
Finding the scale
If you know a drawing length and the matching real length, you can find the scale. Put both in the same unit, write drawing : real, and simplify so the first number is $1$.
In the model of a famous communications tower, the height of the replica is $85$ in. If we know that the real height of the tower is $255$ feet, determine the simplified…
In the model of a famous communications tower, the height of the replica is $85$ in. If we know that the real height of the tower is $255$ feet, determine the simplified numerical scale at which the model has been built.
Bar scales
Many maps print a bar scale (graphic scale) that shows how far a segment on the map is in real life. It gives you a unit rate: so many miles for each inch.
You observe a graphic scale on a hiking map where a $1\text{ in}$ line segment is equivalent to $500\text{ meters}$ in reality. If the route you want to do measures…
You observe a graphic scale on a hiking map where a $1\text{ in}$ line segment is equivalent to $500\text{ meters}$ in reality. If the route you want to do measures $12\text{ in}$ on the map, how many miles will you walk in total?
A road map has a graphic scale in which a $2$ in line segment represents a real distance of $15$ mi. If the distance between two cities on that map is $9,4$ in, what is…
A road map has a graphic scale in which a $2$ in line segment represents a real distance of $15$ mi. If the distance between two cities on that map is $9,4$ in, what is the real distance in miles between both cities?
Enlargement scales
When the real object is tiny, the drawing is bigger than the object. The scale is written $n:1$. A scale of $5:1$ means the drawing is $5$ times as large as the real thing.
A biologist photographs a small queen ant using an enlargement scale of $5:1$. If in the photograph the ant measures $45\text{ in}$ in length, what is its real size in…
A biologist photographs a small queen ant using an enlargement scale of $5:1$. If in the photograph the ant measures $45\text{ in}$ in length, what is its real size in inches?
A biologist observes an ant through a microscope. In his technical drawing, the ant has a length of $12$ in. If the scale of the drawing is an enlargement of $25:1$,…
A biologist observes an ant through a microscope. In his technical drawing, the ant has a length of $12$ in. If the scale of the drawing is an enlargement of $25:1$, what is the real length of the ant expressed in inches?
Scale and area
A scale of $1:k$ multiplies lengths by $k$. Area is length times length, so areas are multiplied by $k^2$.
Area units convert with squares too: $1\text{ ft}^2 = 12 \times 12 = 144\text{ in}^2$, and $1\text{ m}^2 = 100 \times 100 = 10{,}000\text{ cm}^2$.
When a problem asks how many tiles cover a floor, divide the floor area by the area of one tile, in the same unit.
The plan of a municipal soccer field is drawn to a scale of $1:400$. On the plan, the area occupied by the grass is a rectangle $25$ in long by $15$ in wide. Calculate…
The plan of a municipal soccer field is drawn to a scale of $1:400$. On the plan, the area occupied by the grass is a rectangle $25$ in long by $15$ in wide. Calculate the real area of the soccer field in square feet.
You are designing your own "Gaming Room" on a plan with a scale of $1:40$. In the drawing, the room has an area of $75\text{ in}^2$. You have seen a professional desk…
You are designing your own "Gaming Room" on a plan with a scale of $1:40$. In the drawing, the room has an area of $75\text{ in}^2$. You have seen a professional desk that occupies a real area of $1,28\text{ ft}^2$ and a chair that occupies $0,6\text{ ft}^2$.
If you decide to place the desk and two identical chairs, what real area will be left free in the room? Express the result in square feet ($\text{ft}^2$).
The plan of a municipal swimming pool is drawn to a scale such that one of its walls, which measures $25\text{ meters}$ in reality, is represented by a $12,5\text{ cm}$…
The plan of a municipal swimming pool is drawn to a scale such that one of its walls, which measures $25\text{ meters}$ in reality, is represented by a $12,5\text{ cm}$ line segment. On the plan, the bottom of the pool has an area of $120\text{ cm}^2$.
They want to cover the real bottom with square tiles of side $20\text{ cm}$. How many tiles will they need to buy?
Scale and volume
Volume has three dimensions: length, width, and height. Each is multiplied by $k$, so volumes are multiplied by $k^3$.
An architect has built a model of a new auditorium at a scale of $1:250$. The total volume of the model is $0,64\text{ in}^3$. Knowing that the density of the real…
An architect has built a model of a new auditorium at a scale of $1:250$. The total volume of the model is $0,64\text{ in}^3$.
Knowing that the density of the real materials is different, but the volume remains proportional to the cube of the scale, calculate the real volume of the auditorium in cubic feet ($\text{ft}^3$).
(Note: Remember that if the length scale is $k$, the volume scale is $k^3$).
Changing the scale
Sometimes you must move from one scale to another. For example, you may shrink a map to print it, or zoom a map app in or out. The real distance does not change, so use it as a bridge between the two maps.
When you photocopy a map at $p\%$, every length on the page is multiplied by $\frac{p}{100}$. The scale factor $k$ is then divided by $\frac{p}{100}$.
Zooming works the same way. When you zoom a map out, features shrink on the screen, so $k$ gets bigger.
You have two different maps of the same area. On Map A, the distance between two towns is $6$ in and the scale is $1:50.000$. On Map B, the distance between those same…
You have two different maps of the same area. On Map A, the distance between two towns is $6$ in and the scale is $1:50.000$. On Map B, the distance between those same towns is $15$ in. Calculate what the numerical scale of Map B is.
A photocopy has been made of an original plan that was at a scale of $1:200$. Due to a configuration error, the photocopy has come out reduced to $75\%$ of its original…
A photocopy has been made of an original plan that was at a scale of $1:200$. Due to a configuration error, the photocopy has come out reduced to $75\%$ of its original size.
On the photocopy, the living room of the house measures $6\text{ in}$ long.
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What is the real scale of the photocopy with respect to reality?
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What is the real length of the living room in feet?
A map app 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 subway stations is $6\text{ in}$.…
A map app 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 subway stations is $6\text{ in}$. If the user zooms out until the scale is $1:50.000$ (level B):
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What will the distance in inches be between those same stations on the screen?
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If the phone screen has an area of $80\text{ in}^2$, how many real square miles of urban area are being viewed at level B?
Putting it all together
Longer problems combine scale, unit conversions, and speed. Use $\text{distance} = \text{speed} \times \text{time}$, so $\text{time} = \text{distance} \div \text{speed}$. Keep the order: scale first, then units, then speed.
On a road map of the US with a scale of $1:2.500.000$, the distance measured with a ruler between Dallas and Houston is $14,2\text{ in}$. A delivery truck leaves Dallas…
On a road map of the US with a scale of $1:2.500.000$, the distance measured with a ruler between Dallas and Houston is $14,2\text{ in}$. A delivery truck leaves Dallas at $08:00\text{ h}$ and travels towards Houston at an average speed of $90\text{ mph}$.
What time is it expected to arrive at its destination if it stops for half an hour halfway to rest?
An old map has lost its numerical scale indication. However, it shows a graphic scale in which a line segment measuring $3,5\text{ in}$ on the paper represents a real…
An old map has lost its numerical scale indication. However, it shows a graphic scale in which a line segment measuring $3,5\text{ in}$ on the paper represents a real 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 $12\text{ in}$, how many hours will it take a cyclist to go from one to the other at a constant speed of $24\text{ mph}$?
A forest surveillance drone flies a rectangular route over a nature reserve. In the control center, technicians view the route on a digital screen where the scale is…
A forest surveillance drone flies a rectangular route over a nature reserve. In the control center, technicians view the route on a digital screen where the scale is $1:50.000$. On this screen, the rectangle measures $12\text{ in}$ long and $8\text{ in}$ wide. However, for the printed report, they must reduce the map so that the length of the route measures only $4\text{ in}$.
Calculate:
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The real area of the nature reserve in square miles ($\text{mi}^2$).
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The numerical scale that the map in the printed report must have.
Back to the start: Emma's $5$ in. on a $1:250{,}000$ map is $5 \times 250{,}000 = 1{,}250{,}000$ in. Dividing by $63{,}360$ gives about $19.7$ mi. At highway speed, that is a drive of about $20$ minutes.