Measuring Distances - Triangulation

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An Explorer lesson from Nancy Michal at Brea Junior High School.

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Title: Measuring Distances with Triangulation

Author: Nancy Michal, Brea Junior High School, Brea, CA

Theme: Scale and Structure

Grade Level: 7-12

Group size: 2-4 students

Setting: Classroom and outdoors

Approximate Time Required: 2 class periods; can be extended to 3 if you wish to use trigonometry.

Objectives: Students will:

1. Determine the distance to the object by sighting a distant object from 2 different locations and knowing the distance between those locations (parallax).
2. Use trigonometry to determine an unknown distance. (optional)

Major Concepts:

1. By sighting a distant object from 2 different locations and knowing the distance between those locations (called the line of position), we can determine the distance to the object.
2. A carefully drawn scale model can be used to determine an unknown distance.
3. Trigonometry can be used to determine an unknown distance.
4. The farther an object is from an observer, the smaller its parallax.

Science Processes: Measuring, Modeling, Interpreting Data

Vocabulary:

1. Line of Position: The distance between two measured angles (line AB).
2. Parallax: The angle subtended by the far object on the line of position; the apparent shift in position of an object with respect to its background due to a shift in the position of the observer.

Materials Needed (per group):

meter stick, metric ruler, tape, 2 thumbtacks, 2 drinking straws, 2 protractors with small central hole at base, (you can photocopy protractors on card stock instead of using plastic ones).

General Procedures:

1. The procedure is outlined on the student handout.
2. You may want to compile class data for section III, step 2 on the board. Hopefully, the students will discover that the parallax angle gets smaller with increasing distance. This is a good place to discuss the use of parallax to measure distances to stars. Since the angle decreases with distance, this method can only be used for nearby stars.
3. One way to increase the use of parallax is to increase the line of position (section IV on the student handout). Astronomers use the diameter of Earth's orbit (186 million miles) as a line of position. The use of parallax is limited to stars that are closer to Earth than 300 light years.

General Information for Teachers:

1. Assembly of materials can be done ahead of time by a student aide. This not only saves time, but allows you to check the placement of the protractors.

2. Though no worksheet is included, you may wish to make a class set of the procedure. Students can make a data table on their own paper, similar to this:

   Object  Angle 1  Angle 2  Line of Position  Meas. Dist.  Calc. Dist.

   1.
   2.
   3.
   4.
   5.
         Scale drawings can go on the same paper.

3. Caution students to avoid rotating the measurer when they raise it to eye level during the outside portion of this activity.

Reference:

Astronomy Made Simple, Meir H. Degani, Doubleday Made Simple Books, 1976, pp. 82-84

I. Assembly of Materials

1. Push one of the thumbtacks through the sticky side of a piece of tape. The tape should be long enough to wrap around the meter stick.
2. Position the tape with the thumbtack over the 10 cm mark on the meter stick; secure it in place so that the pointed end of the tack is sticking up.
3. Repeat the procedure with the second tack, positioning it over the 10 cm mark of the metric ruler.
4. Place a protractor over each tack. Position the protractor so its base is parallel to the edge of the meter stick/ruler. Tape the protractor in place.
5. Place a straw on each tack. You will sight the object to be measured through the straw.

II. Using the Measuring Device

1. Place the meter stick and ruler end to end on a flat surface. You now have a line of position that is one meter long.
2. Move the left straw so that it is at a right angle to the meter stick. Find a distant object in the classroom. Without moving the meter stick or ruler, move the right straw until you can see the same object through it.
3. Record the measurements of your 2 angles (one is 90 degrees) and your line of position (it should be 1 meter).
4. If possible, measure the distance to the object. You can then compare this distance to the distance obtained by triangulation.
5. On a piece of paper, draw a scale model of your measurements. Use 10 cm to represent your 1 meter line of position; draw this line near the bottom of your paper. Using a protractor, construct the 2 angles you measured at the left and right ends of the line of position.
6. Extend the sides of the triangle until they meet. The angle formed at the top of the triangle is called the parallax. What is its measurement? (Remember that the sum of the three angles of a triangle is 180 degrees.)
7. To find the distance to the object, measure the line between the right angle and the parallax angle. Use your scale (10 cm = 1 m) to convert this to an actual distance.
8. How does your measured distance (from step 4) compare to the distance determined in step 7? What are some sources of error?

III. Comparing Near to Far Objects

1. Move your distance measurer closer to the object Your were measuring. Repeat the steps in Part II.
2. As the distance to the object increases, what happens to the parallax? Compare your results with those of others in your class to see if they are consistent.

IV. Increasing the Line of Position

1. To measure the distance to a much farther object, you will need a longer line of position. Your teacher will direct you to a location outside in which to conduct your work.
2. Move the meter sticks and ruler farther apart measure the length of your new line of position. (Since the protractors are already 1 meter apart, a space of nine meters between the meter stick and ruler will give you a 10 m line of position.)
3. Since the meter sticks are at ground level, you need to lift them in order to sight through the straw. BE CAREFUL TO AVOID ROTATING THE METER STICK AS YOU RAISE IT!

V. Using Trigonometry

1. So far, you have used scale drawings to find the distance to the measured object. There is another method, using a branch of mathematics called trigonometry. Trigonometry enables us to find unknown parts of triangles. The trig function you need for this exercise is called the tangent.
2. The tangent of an angle is the ratio of the side opposite the angle divided by the side adjacent to the angle. In your measurements, you know the adjacent side (it is your line of position) and you want to find the opposite side (the distance to the object). Using a trig table or calculator with a tangent function, you can set up a ratio and solve for the distance to the object. An example follows:
   

   Suppose you had a right triangle with a base of 10 m, and a side angle of 70 degrees, what is its height (X) ?

   Tan = opp / adj
   Tan 70 = 2.7475 (from a trig. table or calculator)
   2.7475 = X / 10.0 m
   X = 27.5 m

3. Use the tangent function to calculate distances to the objects you measured. Compare the results with those obtained from the scale drawings.