Learning Math: Measurement
Indirect Measurement and Trigonometry Part B: Measuring Heights of Tall Objects (35 minutes)
Session 5, Part B
Measuring With Shadows
What other methods exist for measuring indirectly? One such method is to use shadows. For tall objects that are difficult to measure directly, such as skyscrapers and giant redwood trees, shadows can be very useful, as they lie on the ground and are fairly easy to measure. The “shadow” method also relies on similar triangles.
- Go outside on a sunny day with a friend, and find a tall lamppost that is casting a shadow. (If you can’t find a lamppost, use another tall object.) You need to be able to measure the length of the shadow, so be wary of shadows that fall onto roadways. A large parking lot or field is ideal. Measure the length of the shadow of the lamppost. Make sure that the lamppost is on flat, level ground, since the lamppost and the shadow should be perpendicular to each other. Record the measurement.
- Now measure the shadow your friend casts on level ground.
- Make a sketch of the lamppost and shadow and label what you know. Also draw a right triangle that shows your friend and his or her shadow as the legs of another triangle. Again, label what you know. Your sketch should look something like this:
Why are the two triangles (lamppost/shadow and friend/shadow) similar? Think about the angles in each of the triangles. How were those angles formed?
Do the sunbeams form an angle? Also, remember that if two of the angles in both triangles are congruent, the triangles must be similar.
If you know that the triangles are similar, how can you find the height of the lamppost?
Determine the height of the lamppost. Discuss the different proportions that you might use to calculate its height.
Take It Further
- Why is the height of the lamppost a derived measure (i.e., measured indirectly)?
- Let’s assume that your measurements were accurate to the nearest 0.25 m. Use proportions to calculate an upper and lower limit on the height of the lamppost.
- What do you think is the best value for the height of the lamppost?
Imagine that you have a tall tree in your yard that needs to be cut down. You want to make sure that the tree won’t hit your house when it falls. How might you approximate the height of the tree?
In this segment, Katy and Lombi use the method described in this section to find out the height of a tree in the schoolyard. They set up a similar triangle by measuring the length of the shadow of a meterstick.
What are some of the advantages and disadvantages of using a shadow to measure lengths?
You can find this segment on the session video approximately 8 minutes and 28 seconds after the Annenberg Media logo.
This activity should be conducted on level ground. It is important that the object of interest (e.g., a tree or person) and its shadow are perpendicular to each other so that the angle formed in both cases is 90 degrees.
It is also important that you measure the lengths of the shadows at the same time of day. The rays of the Sun hit the tops of these objects at a specific angle, depending on the time of day, and in order for the triangles to be similar, the angles must be congruent.
Note that this method only works with sunlight, and won’t work with lamplight. Rays of sunlight are, for our purposes, parallel. Lamplight rays come from a point source, so they radiate. The same object will cast a different length shadow depending on its distance from the point source.
If you are working in a group, discuss the similarities and differences between measuring shadows and using a transit to find an indirect measure. You might want to consider two important facts:
- Similar triangles must have congruent angles. (How do we guarantee this in both cases?)
- The corresponding sides must be in proportion. (How does this occur in both cases?)
Answers will vary.
The triangles are similar because they both have two equal angles. One equal angle is the right angle formed by the lamppost/friend and the flat ground. The other equal angle is the angle formed by the shadow of each object and the Sun’s rays (the Sun’s rays are parallel lines that strike the ground at the same angle for either shadow).
We can take a known measurement for each object (the length of the shadow) to establish the scale factor by setting up a ratio (AB/DE). Then we multiply the height of the person (EF) by the scale factor to get the height of the lamppost (BC). This is equivalent to setting up a proportion:
AB/DE = BC/EF
Or, using cross multiplication:
BC = (AB • EF)/DE
Answers will vary. You could use the proportion from Problem B3, or alternatively, you could set up a different proportion, which would yield the same result:
AB/BC = DE/EF
Or, using cross multiplication to solve for BC:
BC = (AB • EF)/DE
Both proportions will yield the same result.
- It is a derived measure since it is determined by calculations on other measures.
- Answers will vary depending upon the actual height of the person and the lengths of the shadows. The upper limit will use the maximum lengths for the person and for the lamppost’s shadow, and the minimum length for the person’s shadow. The lower limit will use the opposites. Assuming that you could know the accurate measure for each of these lengths, the upper limit would be BC = ((AB + 0.25) • (EF + 0.25))/(DE – 0.25). (Each of these amounts (AB, EF, DE is the absolute height of the object.)
The lower limit would be BC = ((AB – 0.25) • (EF – 0.25))/(DE + 0.25).
For more information on accuracy, go to Session 2, Part C.
c. The best value for the height of the lamppost might be the average of these two limits, since it gives us a reasonable estimate that is close to either limit.
You could use the methods presented in this part to take a derived measure of the height of the tree.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.