by Joe Cuoio

This page is meant to help you work Dr. Peterson's map scale material. This is an area many, if not most, cartography students have trouble with. Dr. Peterson is consistent with scale, as it will be a major topic on both exams. If scale can be mastered, it will make for a fairly easy section on an otherwise difficult test.

First, some things that are best commited to memory before you take the exam or do scale homework. 1. 1 mile = 5,280 feet = 63,360 inches 2. 1 kilometer = 1,000 meters = 100,000 centimeters 3. 1 mile = 1.6 kilometers 4. 1 inch = 2.54 centimeters 5. 1 square mile = 640 acres 6. 1 hectare = 2.471 acres = 10,000 square meters To take an exam without a working knowledge of these standards would be foolish. Also, it is important to be able to state these equalities more than one way. For instance, 1 inch = 2.54 cm, but what is 1 cm in inches? Well, 1" = 2.54cm so (1/2.54 = 2.54/2.54 (doing the same math operation to both sides of the equation)) results in 1 cm = .394 inch. (1 inch divided by 2.54 = .394) Or, if 1 mile = 63,360 inches, then how many inches is 2.7 miles? Easy- just take 2.7(miles) X 63,360 (inches in a mile) = 171072 .(inches in 2.7 miles) If you can readily and accurately translate these standards from one form to another, then you're doing real fine so far.

Types of scale- We've already looked at English verbal scale, where on a map the scale reads 1 inch = x miles, and metric verbal. (1 cm = x km) A third type of scale is the RF. (representative fraction) Often you will look at a map and see an expression like 1:100,000. What this means is on that particular map, 1 map unit = 100,000 of the same units on the Earth. The type of unit is unimportant, as long as the unit types is the same on both sides of the equation. If the RF of a map is 1:100,000, this means 1 inch on the map = 100,000 inches on the Earth, 1 km on the map = 100,000 km on the Earth, and one pig tail on the map = 100,000 pig tails on the Earth.

Some scale problems worked- 1. Two points are 5.7 inches apart on a map, and you know they are 91 miles apart. Your teacher requires you to figure the scale of the map. First, draw a picture- it only takes a few seconds to draw and deposit the info you have on it.

I always ask myself, what equals what? These scale problems are mostly just that- extracting info from the original problem and deciding which pieces are the same. Once this is done, the problem is usually clearer. In this case, we can see that 5.7 inches is the same as 91 miles, on this map. Forcing myself to see and accept this was always the hardest part, but looking at it long enough, it usually becomes obvious. So, 5.7 inches = 91 miles. To get a usable scale, we must reduce the inches part to one inch, as mapmakers do. To get 1 inch, we divide 5.7" by 5.7". Principles of Algebra tell us to do the same thing to both sides of an equation, so 91 miles is also divided by 5.7. (5.7/5.7 = 91/5.7) This reduces to 1 inch = 15.96 miles. This is the English verbal answer. To get the RF, expand 15.96 miles to inches, (15.96 x 63,360 = 1,011,225) and the RF becomes 1:1011225. (Dr. Peterson will usually want answers rounded off reasonably) To get the Metric verbal scale, change the original info to metric. (5.7 x 2.54(number of centimeters in an inch) = 14.48cm) Do the same with the 91 miles. (91 x 1.6(number of kilometers in a mile) = 145.6km) So, 14.48cm = 145.6km. Dividing both sides by 14.48, we get 1cm = 10.06km.

It's also important to be able to work a problem where you're given the scale and then asked to find something else. For instance, the scale of the map is 1 inch = 100 miles, and you are asked to compute how long a bar scale would need to be to represent 150 miles. We already know that 1 inch = 100 miles, and we know that 150 miles is 1.5 times as long as 100 miles. (150/100 = 1.5) We can take 1.5 and multiply it by both sides of our equation, getting 1.5 inches (1 x 1.5 = 1.5) = 150 miles. (1.5 x 100 = 150) So if the scale of the map is 1 inch = 100 miles, and it is, then your bar scale would have to be 1.5 inches long on the map to represent 150 miles.

On a map with a 1:10,000 scale there is a pasture that measures 1.5 inches by 2 inches. Find the area of the pasture in square miles. A small, simple picture of this problem would be a good idea. First, it would be helpful to convert the 1:10,000(inches in this case) to an English verbal that states miles. This is accomplished by 10,000/63360 = .1578. We can now say that the English verbal scale is 1 inch = .1578 miles. This small number is much easier to work with than 1 inch = 10,000 inches. Since we are working with an area instead of a line, we must find the area of the pasture on the map in square inches. This is done by (1.5inches x 2inches = 3 inches square). Then both sides of the RF expression must be squared. The reason for this is that both sides are still linear, which is not the way the rest of the problem is. (now) Doing this (1 x1: .1578 x .1578) gives us an areal scale of 1 sq. in : 0.024909 sq. mi.. This statement now says, "One square inch on the map = .024909 square miles in reality." The hard work in the problem is over, as all we have left to do is take the number of square inches in the pasture (3 sq.") and multiply it by 0.024909. (3 x .024909 = .074727) The pasture represents .074727 of a square mile. If you needed the answer in acres then take (.074727 X 640 (number of acres in a square mile) = 47.82 acres.