Lab Answers

The UTM Metric Coordinate reader is a device to help someone measure lines, areas, directions and UTM Coordinates accurately and (with practice) quickly on a standard 1:24,000 topographic map. Just slide the reader until the lower left corner of the square is directly on the junction of two one-kilometer UTM references lines (shown either by a grid of thin lines or by tickmarks in the margins of the map). Then, read the reader like a graph--the numbers along the bottom of the square measure the UTM easting of the place in hundreds of meters from the north-south kilometer line. The numbers along the left and right edges show the UTM northing. For example, the coordi nates of point A on the map below are about 342,200 mE and 5,727,750 mN. Use your reader to figure:

NOTE: Because of the distortion of copying the UTM reader, your answers may be anywhere from 20-40 meters different than the readings below. Use the answers below as a "guide". Just be sure you know the process involved in determining UTM coordinates and distances between two points.

  1. The easting of point B is => 342,340mE

  2. The northing of point B is => 5,728,680mN

  3. The easting of point C is => 343,260mE

  4. The northing of point C is => 5,728,280mN

  5. The distance in meters eastward from A to B is => 342,200 - 342,340 = -140m

  6. The distance in meters northward from A to B is => 5,727,750 - 5,728,680 = -930m

  7. The straight-line distance in meters from A to B is => the square root of (-140)2 + (-930)2 or 19,600 + 864,900 = 884500. You must take the square root of this number (I don't know how to put a square root symbol online!) The square root of 884,500 is 940.48 meters.

  8. The distance in meters eastward from B to C is => 342,340 - 343,260 = -920m

  9. The meters northward from B to C is => 5,728,680 - 5,728,280 = 400m

  10. The straight-line distance in meters from B to C is => the square root of (-920)2 + (400)2 or 846,400 + 160,000 = 1,006,400. You must take the square root of this number (1,006,400) which is 1003.19 meters.

The Pythagorean Theorem: c2 = a2 + b2

The key to the usefulness of the UTM is the Pythagorean Theorem, the old Plane Geometry rule that says that the square of the length of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two shorter side s. Given this rule and the UTM coordinates of two places, it is easy (for a computer, anyway) to figure the distance between the places.

For example, suppose you were lying in a tent with UTM coordinates 687,221 meters East and 5,308,336 meters North, Zone 15 (the UTM numbers always go in "alphabetical" order, easting-northing-zone, and yes, the UTM system can be accurate enough to describ e the location of something as small as a single tent.) If the coordinates of a lookout tower were 687,398 meters East and 5,308,793 meters North, zone 15, the tower would be 177 meters "false" east of your tent (just subtract the eastings) and 457 meters "false" north. Its distance on a straight line would therefore be the square root of the sum of 177 squared and 457 squared (or 490 meters, according to my hand calculator; check it on yours). As distances increase, the problems with fitting a square grid to the curved earth increase, but at a human scale, the UTM is one of the most useful ways of specifying locations.