In this lesson, students learn how to determine location by triangulation. We describe the process of triangulation and practice finding your location on a worksheet, in the classroom, and outdoors.

Engineers design systems that require precise and known locations, and often use triangulation calculations to do this. Engineers use triangulation with ground data to determine where in space a satellite is located. Accurately determining a satellite's location is important to adjusting its position to keep it on course. Triangulation technology is also used to inform a robot of its current and target locations. Triangulation helps in spatial modeling to determine the area covered by cell phone transmitters, roadway noise zones, voltage maps and river high-water marks. navigation triangulation compass bearing After this lesson, students should be able to: Understand and describe how to use a map and compass to do triangulation and determine their location Understand and describe how to use a map and compass to do triangulation and determine their location. Understand bearing measurements Understand how navigation technology plays an important role in many types of engineering design Ask the students how they would figure out their location if they were hiking along a trail in the middle of Rocky Mountain National Park. (Possible Answers: Look at a map, use a GPS receiver or a compass.) A map gives you information about an area, but it is up to you to figure out how your actual location matches up with what is shown on the map. You can use visual landmarks to help you figure out your approximate location on the map. You can also use a compass to help narrow down where you are in relation to the map. Ask the students to think about how the compass can be used to find your location. (Answer: You will have to make measurements to landmarks.) Imagine that you are out in the wilderness and you come to the top of a ridge. How can you identify the ridge on the topo map? How can you figure out where you are along the ridge? In this lesson, we will explore how to use triangulation to answer these questions. The most basic skill in using a compass is taking a bearing. This tells you what direction (or bearing) you are facing or what direction someplace is, like a mountain, a tree or a building, from where you currently are located. This skill is essential to any activity for which one might use a compass. Therefore, simply put, a bearing is the direction to something measured as an angle reative to north. It increases as you turn toward the east, with north=0 degrees, east=90 degrees, south=180 degrees, and west=270 degrees. Details on using a compass to measure bearing are also discussed in Lesson 5 of the Navigation Unit. In Figure 2, the black dial shows the bearing angle markings. The bearing will only read out correctly if you have the "N" on the dial aligned with the red arrow (the "Red to North Red" arrow) of the compass needle. After discussing the basics of triangulation, we will discuss a little about correcting for magnetic declination. Triangulation is based on finding an unknown location using angle measurements to two known locations. Mathematically, the two known positions define the two vertices (and length of the one side) of a triangle, and the two bearing measurements define two of the angles of the triangle. These three "knowns" determine the size and orientation of the triangle, thus putting the unknown position at the third vertex of the triangle. In practice, triangulation can be done easily on a map. Imagine there are two landmarks, a building and a tower, at known locations (as illustrated in Figure 3). A lost pedestrian, named Fred, measures the bearing to the tower to be 30 degrees and the bearing to the building to be 345 degrees (see Figure 3). To find his position on the map using triangulation, Fred would draw two lines, one through each of the landmarks beginning where he is standing. The line through the tower should be at an angle of 30 degrees relative to north (as shown in Figure 3), and the line through the building should be drawn at an angle of 345 degrees. His location lies at the intersection of these two lines. You can use the same approach to find the location of an unknown object by measuring the bearing to it from two known landmarks. For example, we have observers located at the building and the tower. Each measures the bearing to a car at an unknown location. In this example, the bearing to the car measured from the building is 150 degrees, and the bearing to the car measured from the tower is 260 degrees. To find the location of the car, we mark the building and tower locations on the map. Then, through each location we draw a line along the measured bearing as shown in Figure 4. The car must be located at the intersection of the two bearing lines. You will notice that the bearing measurements can be made at the landmark or at the unknown location. The difference between a bearing from the unknown location to the landmark, and from the landmark to the unknown location is always 180 degrees. That is why you can use the bearing measured at either location to draw the same line. If you are navigating outdoors with the aid of a topo map and a compass, you can apply the triangulation technique to find your location on the map. Using the compass, you would take a bearing measurement to two visible landmarks such as hilltops, radio towers or other noticeable sites. A good strategy for triangulation is described below. First, set the compass up for the bearing to one of the landmarks. To do this, you rotate the circle showing the directions until the correct bearing is shown along the arrow. Now, set the compass down on the map so that the long, straight edge lies on the landmark (see Figure 5). Then, rotate the whole compass until the north mark is pointed up on the map (along the vertical lines), keeping the straight edge on the landmark. The arrow points along the correct bearing. Draw a line along the straight edge. Your position lies on this line. Repeat the same procedure for the second landmark. The point at which the lines intersect should be your location. In reality, there is one more step you have to take to triangulate on a topo map. When we use a compass to find a bearing with respect to north, we rely on the fact that the geomagnetic North Pole and the geographic North Pole are pretty close to each other. It turns out that the geomagnetic North Pole moves around pretty slowly relative to the geographic North Pole. In 2004 it is located approximately at 82.3° North and 113.4° West, (Refer to http://www.geolab.nrcan.gc.ca/geomag/northpole_e.shtml for more information.) This means that a magnetic compass needle does not point exactly towards geographic north. The angle between geographic and magnetic north at a specific location is known as the magnetic declination. Topo maps help us make adjustments for this. On a topographical map, near the legend, you will see a symbol similar to Figure 7. This picture shows the magnetic declination for a map. The sense of the declination describes whether magnetic north is to the east or west of true north. For easterly declinations, a negative sign is shown. For western declinations a plus sign is shown. These signs indicate how to convert from the true bearing shown on the map to the compass bearing that you would need to follow. For example, in Boulder, Colorado in 2004, the declination is 10.4 deg East. This means that the direction of magnetic north is 10.4 deg east of true north. If a landmark on the topo map is at a (true) bearing of 10.4 deg, its magnetic bearing is 0 deg (10.4 - 10.4 deg). So to get there following a magnetic compass you would head off towards 0 deg as shown on your compass. For locations with westerly declination you would add the declination to the true bearing shown on the map to find the magnetic bearing to follow. When you are using a compass and topo map to find location by triangulation you have to do the reverse. If you measure the bearing to an object with your compass, you have to add an easterly declination or subtract a westerly declination before drawing the bearing line on your map. If you are working with a compass and map, there are two types of adjustments you can make to correct for magnetic declination: temporary and permanent. Temporary - Work with true bearing on the map, and correct the compass measurements each time. For example, if you want to find your location on the map and you measure bearing using your compass to two landmarks, first adjust the bearing you measured before drawing the lines on the map. If the declination is easterly, add the declination to your reading before drawing the line. A compass bearing of 92 degrees in Colorado would be drawn as a line at 102 degrees on the topo map (92 plus 10). If the declination is westerly, subtract the declination from your measured compass bearing before drawing the line. Permanent - On some compasses, there is a permanent adjustment eliminating the need to add or subtract the declination. There will be a small screw adjustment on the backside of the compass. Turning the screw will change where the red outline arrow is pointing. To permanently adjust the compass, turn the screw until the "red-to-north-red" arrow points to the degree of declination. For example, if our declination is 15 west, turn the screw until the arrow is pointing 15 degrees to the west of north, or 345 degrees (360 minus 15). The pictures below show the compass before and after the declination has been corrected. In Figure 8, the left compass shows no declination correction. The red arrow is pointing directly north. The right compass shows a declination correction of 20 degrees west. The red arrow is pointing toward 340 degrees, which is 20 degrees west of north. An orienteering compass is made specifically for use with topographical maps. By measuring the bearing to two or more landmarks with the compass, you can use the basic triangulation and magnetic declination techniques to find your position on the topo map. The extension activity found in the Lesson Extension Activities section provides instructions for determining one's position on a topographical map. The ratio of a distance on a map versus the actual distance it represents. The feature of the map that gives important information about the map. Includes the scale, location and landmarks symbols used by the map. Shows which direction is north on the map. The height of a location. At sea level this would be 0 feet. A map that includes elevation information. Usually used for outdoor travel. Lines on a topographical map that show the elevation along that line. An instrument that uses a magnetized metal bar to indicate the direction of the earth's magnetic poles. The direction in degrees that an object is at, such as a mountain or tree. Classroom Triangles Topo Triangulation Topos, Compasses, and Triangles, Oh My! Ask your students this question: "If you wanted to go somewhere you have never been to, how could you find your way there?" (Possible answer: Use a map.) Are all maps the same size? (Answer: No) What tells us the size of a map and how much area it shows? (Answer: The map's scale.) Where on the map do we find the scale and other important information about the map? (Answer: In the legend.) Ask the students that if they were planning a trip outside, how could a topographical map help them? (Possible answer: It shows the elevation and other interesting features like mountains, rivers and vegetation.) If you were lost, could you find your location? (Possible answer: Yes, if you had a map and compass.) What is the method that you could use to find your location? (Possible answer: Triangulation) Discussion Question: Solicit, integrate and summarize student responses. Ask the students how they could figure out their location if they were hiking along a trail in the middle of a national park or heavily wooded, mountainous area. (Possible answers: Look at a map, use a GPS receiver or use a compass.) Discussion Question: Ask the students and discuss as a class: Imagine that you are out in the wilderness and you come to the top of a ridge. How can you identify the ridge on the topo map? How can you figure out where you are along the ridge? (Answer: Triangulation, which will be explored further in this lesson.) Toss-A-Question: Students should work in groups of two or three. Ask students to independently think of questions formed as a result of the lesson. Have them write the questions on a half sheet of paper. Have students wad up and toss the paper to a team member who then adds his/her answer idea. If working with three to a group, toss paper to 3rd group member to add her/his answer below the question. After all students have written down ideas, have them toss the paper to another team, who reads the question/answers aloud to the class. Discuss answers with the class. Refer to Lesson Closure section for question ideas. A Map Quest: Students search libraries, their homes, or on the Internet on lesson topic. Have students find topographical maps. Ask them to bring one in and share with the class during the next class period. Have students determine their position on a topographical map. Topo maps can be obtained in advance free of charge from the US Geological Survey or purchased at most stores specializing in outdoor equipment. Determine your declination. Either correct this permanently or just find it to be used later. Find two landmarks on the map that are easily identifiable. Antenna towers or big mountains usually work best (see Figure 9). Find those landmarks outside, and take their bearings with your compass. Write these down on the map, possibly next to the landmark on the map. Make sure and correct for declination. If you permanently adjusted your compass, then this is already corrected for you, but if you only did the temporary adjustment, add or subtract that to your bearings. Do the same for the second landmark. These two lines should intersect. You are (more or less) at the point where they intersect. Now you know where you are! To get a more accurate location, you can measure a bearing to a third landmark. With three landmarks, the bearing lines do not all intersect at a single point, but their intersections should form a small triangle. Your location is within this triangle. Finding Your Way with Map and Compass (FS03501, March 2001). December 31, 2002. U.S. Geological Survey. October 16, 2003. <http://erg.usgs.gov/isb/pubs/factsheets/fs03501.html>.