This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.
A topographic map is a map that shows topography and features found on the earths surface. Like any map it uses symbols to represent these features. Lets look at a section of a topographic map showing the area around Spruce Knob in West Virginia. Spruce Knob is the highest point in West Virginia.
This section of a topographic map illustrates many of the common symbols used on topographic maps. The map is repeated below with many of these symbols labeled.
Some of the more common and important topographic map symbols have been pointed out by the purple arrows. More details are given in the text below.
First lets recognize that map symbols are color coded. Symbols in green indicate vegetation, symbols in blue represent water, brown is used for topographic symbols, man made features are shown in black or red. Lets look at the symbols labeled in the map above:
Contour lines are lines that indicate elevation. These are the lines that show the topography on the map. They are discussed in more detail in the next section. Contour lines are shown in brown. Two types of contour lines are shown. Regular contour lines are the thinner brown lines, index contour lines are the thicker brown lines. The numbers written in brown along the contour lines indicate elevation of the line. For this map elevation is in feet above sea level.
Forests and Clearings
Forested areas are represented by areas shaded green; for Spruce Knob this means most of the area. Areas that are not forested are left unshaded (white). Note that not all topographic maps show forests. Also note that this information is not always up to date or accurate. I have struggled to walk across densely wooded areas in places that have been mapped as "clearings".
Streams and other water features are shown in blue.
Roads and Trails
Man made features are shown in black or red. Trails are represented as thin single dashed lines. Roads are represented as double lines or thicker red lines. A series of symbols are used roads to indicate road quality from double dashed lines for dirt roads to thick red lines for major highways. In the case of the Spruce Knob area we have two types of road, the thin double black lines and the thin dashed double lines.
Like other man made features buildings are shown in black. Solid squares usually indicate buildings that would be inhabited by people (i.e. a house), hollow shapes usually indicate uninhabited buildings (for example, a barn) (Note this may not hold for maps in the future because it is not possible to determine what a building is used for from the aerial photos used to make the maps). Other man made features shown in black on our example include the lookout tower on at the summit of Spruce Knob and the radio tower. Though not seen on our map, larger buildings, like factories, are shown by larger shapes that outline shape of the building, and cities with closely spaced houses are shaded pink instead of showing individual houses.
Even though these are not physical features you can see on the ground, boundaries are shown on topographic maps by black or red lines. Boundaries are usually represented by broken lines (combinations of dots and dashes of different sizes). Different patterns are used for different types of boundaries (i.e., state, county, city, etc). On our example the boundary that is shown marks the edge of a National Forest.
Bench marks indicate places where the elevation has actually been surveyed. These locations are indicated on the map by a triangle if a marker has been placed in the ground, or an "x" if not marker was left behind. Near either symbol are the letters "BM" and a number which represents the elevation of that particular location. Bench marks are shown in black on topographic maps.
Contour lines are lines drawn on a map connecting points of equal elevation. If you walk along a contour line you neither gain nor lose elevation.
Picture walking along a beach exactly where the water meets the land (ignoring tides and waves for this example). The water surface marks an elevation we call sea level, or zero. As you walk along the shore your elevation will remain the same, you will be following a contour line. If you stray from the shoreline and start walking into the ocean, the elevation of the ground (in this case the seafloor) is below sea level. If you stray the other direction and walk up the beach your elevation will be above sea level
(See diagram at right).
The contour line represented by the shoreline separates areas that have elevations above sea level from those that have elevations below sea level. We refer to contour lines in terms of their elevation above or below sea level. In this example the shoreline would be the zero contour line (it could be 0 ft., 0 m, or something else depending on the units we were using for elevation).
Contour lines are useful because they allow us to show the shape of the land surface (topography) on a map. The two diagrams below illustrate the same island. The diagram on the left is a view from the side (cross profile view) such as you would see from a ship offshore. The diagram at right is a view from above (map view) such as you would see from an airplane flying over the island.
The shape of the island is shown by location shoreline on the map. Remember this shore line is a contour line. It separates areas that are above sea level from those that are below sea level. The shoreline itself is right at zero so we will call it the 0 ft. contour line (we could use m., cm. in., or any other measurement for elevation).
The shape of the island is more complicated than the outline of the shoreline shown on the map above. From the profile it is clear that the islands topography varies (that is some parts are higher than others). This is not obvious on map with just one contour line. But contour lines can have elevations other than sea level. We can picture this by pretending that we can change the depth of the ocean. The diagram below shows an island that is getting flooded as we raise the water level 10 ft above the original sea level.
The new island is obviously smaller than the original island. All of the land that was less than 10 ft. above the original sea level is now under water. Only land where the elevation was greater than 10 ft. above sea level remains out of the water. The new shoreline of the island is a contour line because all of the points along this line have the same elevation, but the elevation of this contour line is 10 ft above the elevation of the original shoreline. We repeat these processes in the two diagrams below. By raising water levels to 20 ft and 30 ft above the original see level we can find the location of the 20ft and 30 ft contour lines. Notice our islands get smaller and smaller.
Fortunately we do not really have to flood the world to make contour lines. Unlike shorelines, contour lines are imaginary. They just exist on maps. If we take each of the shorelines from the maps above and draw them on the same map we will get a topographic map (see map below). Taken all together the contour lines supply us with much information on the topography of the island. From the map (and the profile) we can see that this island has two "high" points. The highest point is above 30 ft elevation (inside the 30 ft contour line). The second high point is above 20 ft in elevation, but does not reach 30 ft. These high points are at the ends of a ridge that runs the length of the island where elevations are above 10 ft. Lower elevations, between the 10 ft contour and sea level surround this ridge.
With practice we can picture topography by looking at the map even without the cross profile. That is the power of topographic maps.
A common use for a topographic map is to determine the elevation at a specified locality. The map below is an enlargement of the map of the island from above. Each of the letters from A to E represent locations for which we wish to determine elevation. Use the map and determine (or estimate) the elevation of each of the 5 points. (Assume elevations are given in feet)
Point A = 0 ft
Point A sits right on the 0 ft contour line. Since all points on this line have an elevation of 0 ft, the elevation of point A is zero.
Point B = 10 ft.
Point B sits right on the 10 ft contour line. Since all points on this line have an elevation of 10 ft, the elevation of point B is 10 ft.
Point C ~ 15 ft.
Point C does not sit directly on a contour line so we can not determine the elevation precisely. We do know that point C is between the 10ft and 20 ft contour lines so its elevation must be greater than 10 ft and less than 20 ft. Because point C is midway between these contour lines we can estimate the elevation is about 15 feet (Note this assumes that the slope is constant between the two contour lines, this may not be the case).
Point D ~ 25 ft.
We are even less sure of the elevation of point D than point C. Point D is inside the 20 ft. contour line indicating its elevation is above 20 ft. Its elevation has to be less than 30 ft. because there is no 30 ft. contour line shown. But how much less? There is no way to tell. The elevation could be 21 ft, or it could be 29 ft. There is now way to tell from the map. (An eight foot difference in elevation doesn't seem like much, but remember these numbers are just an example. If the contour lines were spaced at 100 ft intervals instead of 10 ft., the difference would be a more significant 80 ft.)
Point E ~ 8 ft.
Just as with point C above, we need to estimate the elevation of point E somewhere between the 0 ft and 10 ft contour lines it lies in between. Because this point is closer to the 10 ft line than the 0 ft. line we estimate an elevation closer to 10. In this case 8 ft. seems reasonable. Again this estimation makes the assumption of a constant slope between these two contour lines.
CONTOUR INTERVAL and INDEX CONTOURS
Contour lines can be drawn for any elevation, but to simplify things only lines for certain elevations are drawn on a topographic map. These elevations are chosen to be evenly spaced vertically. This vertical spacing is referred to as the contour interval. For example the maps above used a 10 ft contour interval. Each the contour line was a multiple of 10 ft. (i.e. 0, 10, 20, 30). Other common intervals seen on topographic maps are 20 ft (0, 20, 40, 60, etc), 40 ft (0, 40, 80, 120, etc), 80 ft (0, 80, 160, 220, etc), and 100ft (0, 100, 200, 300, etc). The contour interval chosen for a map depends on the topography in the mapped area. In areas with high relief the contour interval is usually larger to prevent the map from having too many contour lines, which would make the map difficult to read.
The contour interval is constant for each map. It will be noted on the margin of the map. You can also determine the contour interval by looking at how many contour lines are between labeled contours.
Unlike the simple topographic map used above, real topographic maps have many contour lines. It is not possible to label the elevation of each contour line. To make the map easier to read every fifth contour line vertically is an index contour. Index contours are shown by darker brown lines on the map. These are the contour lines that are usually labeled.
The example at right is a section of a topographic map. The brown lines are the contour lines. The thin lines are the normal contours; the thick brown lines are the index contours. Notice that elevations are only marked on the thick lines.
Because we only have a piece of the topographic map we can not look at the margin to find the contour interval. But since we know the elevation of the two index contours we can calculate the interval ourselves. The difference in elevation between the two index contours (800 - 700) is 100. We cross five lines as we go from the 700 line to the 800 line (note we don't include the line we start on but we do include the line we finish on). Therefore we divide the elevation difference (100) by the number of lines (5) we will get the contour interval. In this case it is 20. We can check ourselves by counting up by 20 for each contour from the 700 line. We should reach 800 when we cross the 800 line.
One piece of important information we can not determine from the contour lines on this map is the units of elevation. Is the elevation in feet, meters, or something else? There is a big difference between an elevation change of 100 ft. and 100 m (328 ft). The units of the contour lines can be found in the margin of the map. Most topographic maps in the United States use feet for elevation, but it is important to check because some do you meters.
Once we know how to determine the elevation of the unmarked contour lines we should be able determine or at least estimate the elevation of any point on the map.
Using the map below estimate the elevation of the points marked with letters
Point A = 700
An easy one. Just follow along the index contour from point A until you find a marked elevation. On real maps this may not be this easy. You may have to follow the index contour a long distance to find a label.
Point B = 740
This contour line is not labeled. But we can see it is between the 700 and 800 contour line. From above we know the contour interval is 20 so if we count up two contour lines (40) from 700 we reach 740.
Point C ~ 770
Point c is not directly on a contour line. But by counting up from 700 we can see it lies between the 760 and 780 contour lines. Because it is in the middle of the two we can estimate its elevation as 770.
Point D = 820
Point D is outside the interval between the two measured contours. While it may seem obvious that it is 20 above the 800 contour, how do we know the slope hasn't changed and the elevation has started to back down? We can tell because if the slope stated back down we would need to repeat the 800 contour. Because the contour under point D is not an index contour it can not be the 800 contour, so must be 820.
DETERMINING CONTOUR INTERVALS
Most contour lines on topographic maps are not labeled with elevations. Instead the reader of the map needs to be able to figure out the elevation by using the labeled contour lines and the contour interval (see previous page for explanation). On most maps determining contour interval is easy, just look in the margin of the map and find where the contour interval is printed (i.e. Contour Interval 20 ft).
For the maps on this web site, however, the contour interval is not listed because we only parts of topographic maps, not the whole map which would include the margin notes. However we usually don't need to be given the contour interval. We can calculate from the labeled contours on the map as is done below.
This method works if we don't have any topographical complications, areas where the elevation is not consistently increasing or consistently decreasing. With practice these areas can usually be easily recognized. Also this method does not tell the units for the contour interval. In the United States most topographic maps, but not all, use feet for elevation, however it is best to check the margin of the map to be sure.
Lets go back to the Spruce Knob area and practice reading elevations. On the map below are 10 squares labeled A through J.? Estimate the elevation for the point marked by each square (make sure to use the point under the square, not under the letter). Compare your answers to the answers below. Recall that we determined the contour interval on the previous page.
ELEVATION of Points:
A. 4400 ft Point A sits right on a labeled index contour. Just follow along the contour line until you reach the label
B. 4720 ft Point B sits on a contour line, but it is not an index contour and its elevation is not labeled. First lets look for a nearby index contour. There is one to the south and east of point B. This contour is labeled as 4600 ft. Next we need to determine if point B is above or below this index contour. Notice that is we keep going to the southeast we find contour lines of lower elevations (i.e. 3800 ft.). This means as we move away from 4600 ft. contour line toward point B we are going up hill. So point B is above 4600 ft. Count the contour lines from 4600 ft to point B, there are three. Each contour line is 40 ft. (from our previous discussion of the contour interval) so point B is 120 ft. above 4600 ft that is it is 4720 ft.
C. 4236 ft Point C sits right on a labeled bench mark so its elevation is already written on the map.
D. 4360 ft. Point D is on an unlabeled contour line. From our discussion of point B above, you can see that point D is on the slope below Spruce Knob. Just above point D is an index contour. If we trace along this contour line we see its elevation is 4400 ft. Since point D is the next contour line down hill it is 40 ft lower.
E 3800 ft. Point E is on an index contour. Follow along this contour line until you come to the 3800 label.
F. ~4780 ft. Point E does not sit on a contour line so we can only estimate its elevation. The point is circled by several contour lines indicating it is a hill top (see the later discussion of depression contours to see why we know this is a hill). First lets figure out the elevation of the contour line that circles point F. Starting from the nearest index contour line (4600 ft) we count up by 40 for the four contour lines. This gives us 4760 ft (4600ft + 40 ft. x 4). Because point F is inside this contour line it must have an elevation above 4760 ft., but its elevation must be less than 4800 ft, otherwise there would be a 4800 contour line, which is not there. We don't really know the elevation just that it is between 4760ft. and 4800ft.
G. 4080 ft. In order to determine the elevation of point G we first must recognize it is on the western slope of Spruce Knob. Looking at the index contours we see that point G is between 4400 ft and 4600 ft contours. (It is a good idea to check the elevations by counting by 40 for each of the contour lines between 4400 and 4600. If the numbers do not work out it may mean that the contour lines, and therefore the topography, are more complicated than a simple slope. That is not the case here.) Counting up two contour lines from 4400 ft. gives our elevation of 4080 ft.
H. ~4100 ft. Point H is circled by a contour line indicating it is the top of a small hill. Its elevation is determined the same way we determine the elevation of Point F. Find the index contour below point F (4000 ft) and count up for the two contour lines (4080 ft). Point F is above this elevation but below 4120 ft because this contour line is not present.
I ~3980 ft. Point I is also not on a contour line. It is also not on the top of a hill because a contour line does not encircle it. Instead it is in between to contour lines on the side of a hill. One of the contour lines is the 4000 ft index contour. The other contour is 3960 ft contour (40 ft lower, you can tell it is lower because you are moving toward the stream which is in the bottom of the valley). The elevation of point I is between 3960ft and 4000ft. Since point I is midway between these two contours we can estimate its elevation as midway between 3960 and 4000.
J ~ 3820 ft. The elevation of point J is found the same way as the elevation of point I.
Topographic maps are not just used for determining elevation; they can also be used to help visualize topography. The key is to study the pattern of the contour lines, not just the elevation they represent. One of the most basic topographic observation that can be made is the gradient (or slope) of the ground surface. High (or steep) gradients occur in areas where there is a large change in elevation over a short distance. Low (or gentle) gradients occur where there is little change in elevation over he same distance. Gradients are obviously relative. What would be considered steep in some areas (like Ohio) might be considered gentle in another (like Montana). However we can still compare gradients between different parts of a map.
On a topographic map the amount of elevation change is related to the number of contour lines. Using the same contour interval the more contour lines over the same distance indicates a steeper slope. As a result areas of a map where the contour lines are close together indicate steeper slopes. Areas with widely spaced contour lines are gentle slopes. The map below examples of areas with steep and gentle gradient. Note the difference in contour line spacing between the two areas.
Compare the slope of the west side of Spruce Knob with the slope of the east side. Which side is steeper?
.....The east side. Notice the spacing between the contour lines. Contour lines on the east side of Spruce Knob are closer together than the contour lines on the west side indicating steeper slopes.
Topographic maps are drawn to scale. This means that distances on a map are proportional to distances on the ground. For example, if two cities 20 miles apart are shown 2 inches apart on a map, then any other locations that are two inches apart on the map are also 20 miles apart. This proportion, the map scale, is constant for the map so it holds for any points on the map. In our example the proportion between equivalent distances on the map and on the ground is expressed as a scale of 1 inch = 10 miles, that is 1 inch on the map is equal to 10 miles on the ground. Map scales can be expressed in three forms. We will look at all three.
The simplest form of map scale is a VERBAL SCALE. A verbal scale just states what distance on a map is equal to what distance on the ground, i.e. 1 inch = 10 miles from our example above. Though verbal scales are easy to understand, you usually will not find them printed on topographic maps. Instead our second type of scale is used.
Fractional scales are written as fractions (1/62500) or as ratios (1:62500). Unlike verbal scales, fractional scales do not have units. Instead it is up to the map reader to provide his/her own units. Allowing the reader of the map to choose his/her own units provides more flexibility but it also requires a little more work. Basically the fractional scale needs to turn in to a verbal scale to make it useful.
First lets look at what a fractional scale means. A fractional scale is just the ratio of map distance to the equivalent distance on the ground using the same units for both. It is very important to remember when we start changing a fractional scale to a verbal scale the both map and ground units start the same. The smaller number of the fractional scale is the distance on the map. The larger number in the scale is the distance on the ground.
So if we take our example scale (1:62500) we can choose units we want to measure distance in. Lets chose inches. We can rewrite our fractional scale as a verbal scale:
1 inch on the map = 62500 inches on the ground.
We can do the same thing used with any unit of length. Some examples of verbal scales produced using various units from a 1:62500 fractional scale are given in the table:
UNITS VERBAL SCALE
Inches 1 inch on the map = 62500 inches on the ground.
Feet 1 foot on the map = 62500 feet on the ground
cm 1 cm on the map = 62500 cm on the ground
M 1 m on the map = 62500 m on the ground
Notice the pattern. The numbers are the same, only the units are changed. Note that the same units are used on both sides of each of the verbal scale.
While these verbal scales are perfectly accurate, they are not very convenient. While we may want to measure distance on a map in inches, we rarely want to know the distance on the ground in inches. If someone asks you the distance from Cleveland to Columbus they do not want the answer in inches. Instead we need to convert our verbal scale into more useful units.
Lets take our example (1 inch on the map = 62500 inches on the ground). Measuring map distance in inches is OK, but we need to come up with a better unit for measuring distance on the ground. Lets change 62500 inches into the equivalent in feet (I choose feet because I remember that there are 12 inches in 1 foot). If we multiple 62500 inches by the fraction (1 ft / 12 in) inches in the numerator and denominator cancel leaving an answer in feet. Remember, since 1 ft = 12 inches, multiplying by (1 ft / 12 in) is the same as multiplying by 1. The result of this multiplication gives:
62500 inches x (1 ft / 12 in) = 5208.3 ft
So we can rewrite our verbal scale as 1 inch on the map = 5208.3 feet on the ground.
This is also a perfectly valid verbal scale, but what if we wanted to know the distance in miles instead of feet. We just need to change 5208.3 feet into miles (we could change 62500 inches into miles but I never remember how many inches are in 1 mile). Knowing that there are 5280 feet in a mile:
5208.3 ft x (1 mi/5280 ft) = 0.986 mi.
So our verbal scale would be: 1 inch on the map = 0.986 miles on the ground. For most practical purposes we can round this off to 1 inch on the map ~ 1mile on the ground, making this scale much easier to deal with.
We can do the same type of conversions using metric units. One of the ways to express a fractional scale of 1:62500 as a verbal scale using metric units is 1 cm on the map = 62500 cm on the ground (see table above). As with inches, we really do not want ground distances in cm's. Instead we can convert them into more convent units.
Lets convert our ground distance from cm's into meters. Recall that there are 100 cm in a meter. So:
62500 cm x (1m / 100cm) = 625 m.
So we can write a verbal scale of 1 cm on the map = 625 m on the ground.
What if we want our distance in kilometers (km). We just change 625 m into km by multiplying by (1km/1000m). The result is a verbal scale of 1 cm on the map = 0.625 km on the ground.
So for any fractional scale we can choose the same units to assign to both sides and then convert those units as we see fit to produce a verbal scale. Given all of the possible map scales and all of the possible combination of units that can be used it may seem that scales on topographic maps a very complicated. In fact there are only a few scales commonly used, and each is chosen to allow at least one simple verbal scale. The most common fractional scales on United States topographic maps and equivalent verbal scales are given in the table below.
FRACTIONAL SCALE SIMPLE VERBAL SCALE
1:24000 1 in = 24000 ft
1:62500 1 in ~ 1 mi
1:100000 1 cm = 1 km
1:125000 1 in ~ 2 mi
1:250000 1 in ~ 4 mi
After all this why would anyone in their write mind want to deal with fractional scales. Well, first as the table above shows its not that bad, and second, they allow us to get the most precise measurements off a topographic map. If we are not that concern about being precise we can use the third type of scale, discussed below.
A bar scale is just a line drawn on a map of known ground length. There are usually distances marks along the line. Bar scales allow for quick visual estimation of distance. If more precision is needed just lay the edge of a piece of paper between points on the map you want to know the distance between and mark the points. Shift the paper edge to the bar scale and use the scale like a ruler to measure the map distance.
Bar scales are easy to use, but there is one caution. Look at the typical bar scale drawn below. Note that the left end of the bar is not zero. The total length of this bar is FIVE miles, not four miles. A common error with bar scales is to treat the left end of the line as zero and treat the whole bar as five miles long. Pay attention to where the zero point on the bar actually is when you measure with a bar scale.
In addition to their ease of use, there is one other advantage of a bar scale. If a map is being enlarged or reduced, a bar scale will remain valid if it is enlarged and reduced by the same amount. Fractional and verbal scales will not be valid (unless they are adjusted for the enlargement or reduction, more fun calculations we will not worry about). This is a problem with the maps you are looking at on this web site. The actual scale of the map will vary depending on your computer monitor and its setting. For the maps on this site only bar scales are included since the size of the bar will also change with the size of the map.
Latitude and Longitude
It is important when using topographic maps to have some way to express location. You may want to tell someone where you are (i.e. help we are sinking at this location), or where to go (meet me at this location), or even just what map to look at (look at the map showing this location). In each case you need to be able to express your location as precisely as possible.
There are many systems for expressing location. We will start by looking at one you are already familiar with: latitude and longitude.
Latitude and longitude lines form a grid on the earth's surface. Latitude lines run east to west, longitude lines run north to south. Latitude lines run parallel to the equator and measure the distance north or south of the equator. Values for latitude range from 0° at the equator to 90° N or 90°S at the poles. Longitude lines run parallel to the Prime Meridian (arbitrarily set to run through Greenwich, England) and measure distance east and west of this line. Values of longitude range from zero degrees at the Prime Meridian to 180°E or 180°W.
The basic unit of latitude and longitude is the degree (°), but degrees are a large unit so we often have to deal with subdivisions of a degree. Sometimes we just use a decimal point, such as 35.789°N. This format referred to as decimal degrees. Decimal degrees are often found as an option on Global Position Systems (GPS) or with online topographic maps, but decimal degrees are not used on printed maps. On these topographic maps the latitude and longitude units are expressed in degrees, minutes, and seconds. Each degree is subdivided into 60 minutes ('). Each minute is divided into 60 seconds (''). Note the similarity to units of time which makes these relationships easy to remember. If we are interested in a general location we may just use degrees. For more precision we specify minutes, or even seconds. Note that we always need to specify the larger unit. You can't specify your latitude or longitude with just minutes or seconds. A coordinate such as 25' is meaningless unless the degrees are also given, such as 45° 25'.
The area covered by the quadrangle depends on the spacing of the latitude and longitude lines used in the grid. For maps of roughly the same size closer spaced lines produce maps that cover less area, but show more detail. Lines that are spaced further apart produce maps that cover much larger areas, but are not as detailed. Quadrangles are often referred to by the spacing of these lines. For example we distinguish 7½ minute quadrangles that cover an area of 7½ minutes of latitude by 7½ minutes longitude, from 15 minute quadrangles, which cover an area of 15 minutes latitude by 15 minutes longitude. For standard topographic maps each type of quadrangle is associated with a specific map scale as shown in the table below.
40° 30' N
Point A is in the upper left corner of map so its coordinates are the printed coordinates of this corner. The one thing that needs to be added is the direction notations of each coordinate. They are not printed on the map because it is assumed you can tell what hemisphere you are in. When you are asked for latitude and longitude you must add these letters. It is easy to tell where you are by which direction the numbers for latitude and longitude increase. Latitude increase going north on this map so we are in the northern hemisphere. Longitude increases going to the west, so this map is located west of the Prime Meridian.
40° 25' N
117° 55' W
To determine the location of point B we need to read across to the side of the map (to determine latitude) and up to the top of the map to determine longitude. Point B lines up with labeled tick marks labeled 25' and 55', but we know these numbers are incomplete. looking at the corner of the map we see that the latitude is 40° 25' N (north because of same argument for point A) and the longitude is 117° 55' W.
40° 20' N
117° 50' W
Follow the same procedure as point B above.
40° 27' 30" N
117° 47' 30" W
Point D does not line up directly with tick marks. Instead we need to estimate its location. Point D looks like it is half way between the 25' and 30' marks for latitude and half way between the 45' and 50' marks for longitude. Half way for each of these is 27'30" and 47'30". Remember one half a minute is 30 seconds. Adding the remaining parts of the coordinates as we did above give us the answer.
40° 16' N
117° 52' 30" W
Solved the same as point D above. The only difference is in estimating the minutes for latitude. Point E seems to closer to 15' that to 30' so I have estimated it as 16'. This is only an estimate so the answer can vary, but it should be greater than 15' and less than 17' 30"