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A map's purpose decides its content and scale; a representative fraction turns map measurements into ground sizes and sets the smallest feature a map can show.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to match a map to the question it answers and use its scale to find ground sizes and the detail it can show.
You have read many maps and know they leave things out. This course treats maps as data: a map is a set of choices about what to measure, how to summarize it and how to draw it, and each choice can be checked with numbers. It begins with the first choice, purpose, and the first number, scale.
| Term | What it means |
|---|---|
| Purpose | The question a map answers and the audience it answers it for. |
| Representative fraction | A scale written as a ratio, such as 1:24,000, in any one unit. |
| Large scale | A larger fraction: a small area shown in detail, such as 1:24,000. |
| Small scale | A smaller fraction: a large area shown with little detail, such as 1:1,000,000. |
| Generalization | Simplifying, merging or dropping features the scale cannot show. |
| Quadrangle | A standard U.S. Geological Survey map sheet, often at 1:24,000. |
A map begins with a question and an audience. The question decides what the map must show and what it can leave out; the audience decides how it should look.
Another way: picture
Picture photographing a park from a drone that climbs higher and higher. At first every bench shows; then only the paths; then the park is a green patch among streets. Each height is a different scale, and each answers a different question.
Another way: steps
At different scales, one centimeter on the map stands for very different ground.
| Scale | One map centimeter | Typical use |
|---|---|---|
| 1:24,000 | 240 m | hiking, local planning |
| 1:100,000 | 1 km | county road maps |
| 1:1,000,000 | 10 km | state and regional maps |
The first is the largest scale and the most detailed; the last is the smallest scale and covers the most ground.
The words sound backward to many people. A large-scale map shows a small area; a small-scale map shows a large one. The words describe the fraction: $1/24000$ is a larger number than $1/1000000$.
A world map is the smallest scale of all. A plan of a single building is among the largest.
A flood insurance map shows flood zones and elevations and leaves out restaurants. A transit diagram keeps the order of stations and the connections between lines and drops true distance altogether. A topographic map keeps the shape of the land for hikers and engineers.
None of them is the correct map of a place. Each is correct for its question, and misleading if used for a different one.
At every scale, some features are too small to draw truly. The mapmaker may drop them, merge them into a larger area, straighten a winding line, or replace them with a symbol drawn larger than life.
At $1:24000$, anything smaller than about $12$ m on the ground is below half a millimeter on the page. At $1:1000000$ the limit is about $500$ m, so a whole village becomes a dot.
To fit an area on a page, divide the area's width by the page's width in the same unit. An area $20$ km wide on a page with $40$ cm of room needs a scale of $2000000 \div 40$, which is $1:50000$.
If that scale is too small to show what the question needs, the answer is more sheets, or an inset map that enlarges one part.
A representative fraction works in any unit but breaks if the map is enlarged or shrunk, for example on a screen. A scale bar stretches with the map and stays true.
Digital maps usually show a scale bar that updates as you zoom, because their scale changes continuously.
The U.S. Geological Survey's best-known series is the 7.5-minute quadrangle at $1:24000$, which covers the lower forty-eight states in tens of thousands of sheets. One inch on those maps is $2000$ feet on the ground.
Smaller-scale USGS series at $1:100000$ and $1:250000$ cover larger areas for regional planning, with correspondingly less detail.
Checking an answer. A larger denominator must give a larger ground size for the same map measurement, and a smaller scale must show less detail.
Multiplying by the denominator is allowed because a representative fraction is a ratio of lengths in the same unit, true across a local map. Converting units afterward is allowed because the ratio does not depend on the unit.
Using half a millimeter as a limit is a convention, not a law; it reflects what a reader can see and a printer can draw.
On a world map no single scale is true everywhere, because a round Earth cannot be flattened without stretching. The scale printed on such a map is true only along certain lines.
Local maps, like quadrangles, are small enough that the stretch is negligible, which is why ground distances can be measured from them directly.
A map made for one purpose and used for another can mislead without a single wrong line. A tourist map may shrink distances to make attractions look close; a transit diagram cannot be used to judge walking distance.
Stating the purpose, source and date on the map lets readers judge what it can support, a theme the next lesson develops.
The most common slip is thinking large scale means a large area. Another is mixing units, reading the denominator of $1:24000$ as meters when the map was measured in millimeters.
A third is reading a symbol's size as the feature's size. A fourth is using a map made for one question to answer another.
The Federal Emergency Management Agency publishes Flood Insurance Rate Maps for communities across the United States. Their purpose is narrow and consequential: to show which properties lie in the special flood hazard area, the land with a one percent annual chance of flooding, where federally backed mortgages require flood insurance.
That purpose sets everything about them. They are drawn at large scales so that a single lot can be judged, they show flood zones and base flood elevations rather than shops or parks, and they carry an effective date, because a revised map can move a home in or out of the zone and change a family's insurance bill by thousands of dollars a year.
Used for another purpose, the same map misleads. A home just outside the mapped zone is not safe from floods; the map answers an insurance question, not a guarantee. Many flood claims come from outside the mapped zones, which is why FEMA urges people there to consider insurance too.
The New York City subway map is a schematic diagram. Its purpose is to help riders choose a line, see where lines connect and count stops, so it straightens routes, evens out the spacing of stations and enlarges crowded Manhattan while shrinking the outer boroughs.
In 1972 the city issued a strikingly abstract diagram by the designer Massimo Vignelli, with straight lines at neat angles and a beige Central Park drawn as a square. Riders found it beautiful and confusing: it looked like a map, so they used it to judge walking distances it was never designed to show. The city replaced it in 1979 with a design closer to real geography.
The episode shows the lesson's point. A map's design follows its purpose, and readers bring their own questions. When the two do not match, a map can be accurate for its purpose and still lead people astray.
The phrase large-scale map sounds as if it should cover a large area, but it means the opposite: the fraction is large, so each map unit stands for little ground, and the map shows a small area in detail.
Check with the fraction itself: $1:24000$ is larger than $1:1000000$, so the quadrangle is the large-scale map and the state map the small-scale one.
A symbol is $2$ mm wide on a map at $1:50000$. Multiply.
$2 \times 50000 = 100000\ \text{mm}$
Ground millimeters.
Convert to meters.
$\dfrac{100000}{1000} = 100\ \text{m}$
A thousand millimeters a meter.
Say whether the feature is that big.
$\text{not necessarily}$
Symbols can be enlarged.
Find the true-size limit at this scale.
$0.5 \times 50000 = 25000\ \text{mm} = 25\ \text{m}$
Half a millimeter.
A county is $30$ km wide; the page has $25$ cm of room. Convert the width.
$30 \times 100000 = 3000000\ \text{cm}$
Same unit as the page.
Divide by the page width.
$\dfrac{3000000}{25} = 120000$
The denominator.
Write the scale.
$1:120000$
A representative fraction.
Find the meters per centimeter.
$\dfrac{120000}{100} = 1200\ \text{m}$
Each page centimeter.
Find the smallest true feature.
$0.5 \times 120000\ \text{mm} = 60\ \text{m}$
Smaller things must be generalized.
A park planner needs to show paths $3$ m wide. At $1:24000$, find the true-size limit.
$0.5 \times 24000\ \text{mm} = 12\ \text{m}$
Too coarse for the paths.
Find the limit at $1:5000$.
$0.5 \times 5000\ \text{mm} = 2.5\ \text{m}$
Paths can show.
Choose the scale.
$1:5000$
It keeps what the question needs.
The park is $1.5$ km wide. Find its width on the map.
$\dfrac{150000\ \text{cm}}{5000} = 30\ \text{cm}$
It fits a large sheet.
Say what the other scale is good for.
$\text{showing the park among its neighborhoods}$
A different question.
State the lesson.
$\text{purpose chooses the scale}$
Not the other way around.
Multiply by the denominator.
$4 \times 25000 = 100000\ \text{mm}$
Ground millimeters.
Convert to meters.
$100\ \text{m}$
Divide by one thousand.
Say what it may hide.
On a map at $1:100000$, a building's symbol is $3$ mm wide. How wide a strip of ground does that symbol cover, in meters?
Complete the worked solution: a planner must fit an area $20$ km wide onto a page with $40$ cm of room. Find the scale's denominator and the meters of ground each page centimeter stands for.
Find the scale's denominator.
$\dfrac{\text{area in cm}}{\text{page cm}} =$ d
A kilometer is one hundred thousand centimeters.
Find the meters per page centimeter.
$\dfrac{\text{denominator}}{\text{one hundred}} =$ m
Centimeters to meters.
Say what the scale decides.
$\text{the smallest feature it can show}$
About half a millimeter of page.
Say what to do if detail is too coarse.
$\text{split the area over several sheets}$
A larger scale on each.
Match each question to the map built to answer it.
| a FEMA flood insurance rate map | a schematic transit diagram | a U.S. Geological Survey topographic map | a county choropleth of vote shares | |
|---|---|---|---|---|
| Does this house need flood insurance? | ||||
| Which subway line reaches the museum, and where do I change? | ||||
| How steep is the trail to the summit? | ||||
| What share of each county's votes did each party win? |
Fill in how many meters of ground one centimeter on the map stands for at each scale.
| meters | |
|---|---|
| 1:24,000 (meters per map centimeter) | |
| 1:100,000 (meters per map centimeter) | |
| 1:1,000,000 (meters per map centimeter) |
On a certain map, one millimeter stands for $50$ m of ground. Write the ground length in meters as a function of a length $d$ measured on the map in millimeters.
Answer:
A mapmaker will not draw anything smaller than half a millimeter on the page. At a scale of $1:24000$, how large, in meters, must a feature be on the ground to be drawn at its true size?
Answer: m
Suppose a trail in the Sierra Nevada, California measures $30$ cm on a U.S. Geological Survey quadrangle at $1:24000$. How long is it on the ground, in kilometers?
Answer: km
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
On a certain map, one millimeter stands for $100$ m of ground. Write the ground length in meters as a function of a length $d$ measured on the map in millimeters.
Answer:
You can judge a map's purpose and scale. Explain why a 1:24,000 map is called large scale though it covers a small area.
23. Your turn: a feature is $4$ mm wide on a map at $1:25000$. How wide is its ground strip?, step 3
$\text{an enlarged symbol}$
Drawn big enough to see.