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Measurement via Triangulation These exercises and lesson plans are designed to accompany and enrich the study and discussion of the June 2004 Transit of Venus. Goal: Students use triangulation techniques to determine altitudes of objects. Grade Level: 6-12 Objectives: Construct an altitude locator Use altitude locator to determine angular distance of a building or tree Determine height of building or tree by using tangent tables and formula Construct proportional triangles to determine height of building or tree Subject Area or Standard: Science and Measurement Geometry Materials Needed: Measurement instruments (rulers, meter sticks, tapes, etc.) Protractors Cardboard, paperclip, string, glue Hole punch Drinking straws Paper and pencil Websites: Printable protractor http://www.teachervision.com/lesson-plans/lesson-6197.html Ptolemy's Ptools http://library.thinkquest.org/19029/triangulate.htm Tangent Table Angle in degrees 10 15 20 25 30 35 40 Ratio .176 .268 .364 .466 .577 .700 .839 Angle in degrees 45 50 55 60 65 70 75 Ratio 1.0 1.088 1.176 1.268 1.364 1.466 1.577 Procedures: 1. Students construct altitude locator according to directions: a. Glue protractor to cardboard b. Punch hole at midpoint in ruler side of protractor c. Tie paper clip to string and then thread string through hole in protractor and tie. d. Glue drinking straw along ruler side of protractor above the punched hole. 2. One student stands at base of object (building or tree) and walks away in straight line to an even distance (far enough to be able to site the top of the object through the drinking straw). Ruler side is the top and rest of protractor hangs down. 3. Measure distance from base of object (building or tree) to student with altitude locator. 4. Measure height of student to eye level. 5. Student sites to top of object with altitude locator and finds number of degrees of angle marked by position of string on protractor. 6. Look up tangent ratio from above table and multiply by the distance between the object's base and the siting position. Subtract the eye level height of the student who found the angle. The result is the height of the object. Alternatives: Calculate height using proportionally similar triangles Students obtain the angle by using the altitude locator and they measure the distance from the base of the object to siting position as above. But then students construct a right angle triangle on paper using the same angle of elevation a portion of the distance measure. For example if the distance had been 40 meters, make the base of the triangle 4 centimeters (1/100 size). At one end of the 4 centimeter line draw a line perpendicular (90 degrees) to base. At the other end, draw a line at the angle of elevation. Complete triangle where lines cross. Measure length of right angled side and multiply by the proportion (in this case by 100).