Indirect Measure-Measuring Altitude and Astronomical Bodies.
Geometry
Unit Objective: Students will use different techniques for finding indirect measurements. Although the
activities were written for a Geometry course, they could easily be adapted for other courses.
Activity 1: Making an Astrolabe
Objective: Students will make an astrolabe to use during other activities. History and uses of the
astrolabe will also be discussed.
Wyoming Standards: Sci 3.1-Students examine the nature and history of science.
Materials Needed: Cardboard or manila folder, dark thread, small metal washers, straws, glue, scissors,
tape, hole puncher, copy of astrolabe drawing, printed copy of directions from web site
Time: 25 minutes
Related Links: http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_07.html
Directions: Have the students use the directions to make their astrolabe. This will take 15-20 minutes.
Afterwards discuss the history and uses of the astrolabe. Details from the web site include:
“The astrolabe was invented in Greece either by Hipparchus, a 2nd century B.C. astronomer, or
Apollonius of Perga, a 3rd century B.C. mathematician. For many centuries, it was used by both
astronomers and navigators, and especially by the 15th century explorers who used it to determine
latitude, longitude, and time of day.”
If you would like much more detail about the history and uses you can find information on the Internet.
Activity 2: Measuring Altitude Using an Astrolabe
Objective: The students will use their astrolabe to measure the height of a large object using tangent
ratio.
Wyoming Standards: Math 2.2-Students communicate, using mathematical language, to analyze
properties and determine attributes of 2- and 3- dimensional objects.
Math 2.5-Students connect geometry with other mathematical topics.
Math 3.1-Students apply estimation and measurement using he appropriate
methods and units to solve problems involving length.
Math 3.5-Students solve indirect measurement problems.
Math 4.5-Students connect algebra with other mathematical topics.
Sci 2.2-Students use inquiry to conduct scientific investigations.
Materials Needed: Astrolabe, calculator, long tape measure (100')
Time: 40 minutes
Related Links: http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_08.html
Directions: First have the students use their astrolabe to measure the angle of elevation of different
objects. This will require the students work with a partner. Have one person view the top of a tree or
other tall object using the astrolabe. The other person will read the angle from the side. Have the
students switch places and perform another measurement. Discuss with the students their results and
note the variance between their measurements.
Next make a sketch on the board showing how an astrolabe can be used to measure height or altitude.
h
A
d
Make sure you point out the only distance that is easy to measure is from the person to the object along
the ground (d). This will be the adjacent side of the triangle. The height of the object will be the
opposite side. Since tangent is defined as opposite over adjacent, we can find the missing quantity (the
height).
Also point out that if the person doing the observation has his or her eyeball 1.5 meters above the
ground, then you will have to add 1.5 to your solution. Otherwise the observer would have to lay at
ground level. Before you take the students outside also decide on what units they will use to do their
measurements. Metric will be easier to use but they make need some review on how to read a metric
tape measure. Measuring their desks or their heights using metric would be a good, quick review.
Next take the students outside to measure a couple different objects. It is best to have two or three
objects for them to choose from such as a building, a flag pole, and a tree. Activity 3: Finding Diameter
of Sun and Moon Using Pinhole Viewer
Objective: The students will make indirect measurements using similar triangles and proportions.
Wyoming Standards: Math 1.4-Students will use proportional reasoning to solve problems.
Math 2.5-Students connect geometry with other mathematical topics.
Math 3.3-Students identify and apply scale, ratios, and proportions in solving
measurement problems.
Math 3.5-Students solve indirect measurement problems.
Math 4.5-Students connect algebra with other mathematical topics.
Sci 1.9-Students develop an understanding of scientific content through inquiry.
Sci 2.2-Students use inquiry to conduct scientific investigations.
Materials Needed: Cardboard, white paper, aluminum foil, pin, tape, scissors, ruler, candle, lighter,
printed copy of directions from web site
Time: 50 minutes
Related Links: http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_03.html
Directions: Have the students use the directions to make their pinhole viewer. This will take 10-15
minutes.
Once the students are done show them how the viewer works using a candle and white paper for a
projection screen. It is best to have this set up before the students come to class using a pinhole viewer
you have already made and tried out. Explain how the viewer works including a sketch of the preimage
and image. Also discuss why the image is inverted.
Screen
Preimage
Image
(inverted)
Using this sketch prompt the students to figure out a way to measure the diameter of the sun. They
should be able to figure out there will be two similar triangles. Use the two triangles to set up a
proportion:
(diam of sun)/(dist from Earth to Sun)=(diam of image)/(dist from image to pinhole)
This equation can be rearranged to form the equation used on the web site.
The students can then set up their apparatus to try to measure the diameter of the sun. The actual
distance from the Earth to the sun is 93 million miles or 149.6 million kilometers. When finished the
students can look up the diameter of the sun on the Internet (870,000 mi. or 1.4 million km.).
**WARNING: Make sure your students do NOT look directly at the sun!**