Ask students to tell their classmates about the tallest place they've
ever been. If students remember, have them list the height above sea
level of those locations, and ask them to describe what they saw from
that height. In most cases, students will be able to describe how far
they were able to see to the horizon. For reference, research the
height above sea level of your current location, as well as the height
above sea level of a few nearby attractions. If there is a well-known
mountain nearby, find out the height of its summit. Should students
have difficulty remembering details about places they've been, use the
data about nearby locations to give students a sense of height above
Now ask students to imagine what it would be like to stand on the
top of Mount Everest, which is 29,035 feet above sea level. Ask them,
"How far do you think you'd be able to see to the horizon if you were
standing at the summit of Mount Everest?" Allow students to speculate,
but do not offer opinions about their estimates. Explain that the focus
of this lesson is for them to attempt to answer that question.
Distribute the first page of the Top of the World
activity sheet. Inform students that they should answer questions 1
and 2 fairly quickly; at this point, their guess should be based solely
on a quick estimate, not on extensive research or calculations. Later
in the lesson, they will have an opportunity to develop a general rule
based on data provided.
After all students have completed the first page, conduct a
brief discussion about their answers. However, do not judge or comment
on their estimates. You may allow other students to offer comments, but
you should only say that thinking about this estimate is a precursor to
the activity that they will complete today. Then, divide the class into
pairs or small groups; to half of these groups, distribute the sheet
titled "Algebraic Approach," and to the other half of the groups,
distribute the sheet titled "Geometric Approach." Allow the groups to
work on their respective sheets, and then have groups with different
sheets compare their results. They may be surprised by the results.
After students have completed the activity sheet, conduct a
brief class discussion. Focus this discussion on the following
- Using your formulas, determine how far a person standing at the top
of Mount Everest would be able to see to the horizon. Compare the
results generated by each formula.
- Consider a rocket 1000 miles (or 5,280,000 feet) above the
Earth. Use both of your formulas to determine how far a person in this
rocket would be able to see to the horizon. Do both formulas give the
To conclude the lesson, allow students to explore the Distance to Horizon
applet. This tool allows the user to adjust the height above sea level,
and the distance to the horizon is calculated. The height can be
adjusted to exactly 29,035 feet, the height of Mount Everest. At this
height, the applet shows that the distance to the horizon is
208.8 miles. Students can use this value as a check of their results.
Selected Solutions to the Answer Sheet
Question 4. A power regression (accessible through the
PwrReg feature on TI calculators) gives the function of best fit. The
regression corresponds to the function d = 1.218 · h0.501.
(This result is very similar to a formula used by ship captains who
wished to estimate the distance to a shore. To calculate the distance
to the horizon, they estimated the value of d = (1.5h)0.5.)
Questions 5 and 9. The power function above yields a
result of 208.9 miles, whereas the formula that results from the
geometric approach yields a result of 208.8 miles.
Question 7. The Pythagorean theorem can be used to
generate an equation, but it must be remembered that the distance to
the horizon is measured in miles, while the height above sea level is
measured in feet. Consequently, a conversion needs to be done. Let n = h/5280, which is the height above sea level in miles. Then, the resulting equation is (3963 + n)2 = 39632 + d2.
Question 8. Solving for d then gives d = (7926n + n2)0.5.
Questions for Students
1. How can the distance to the horizon be determined based on the height above sea level?
[The sight line from a certain height to the horizon is tangent to the
Earth and therefore forms a right angle with the radius to the point of
tangency, as shown below.
Consequently, a right triangle is formed, so the Pythagorean theorem
can be applied. It must be known that the approximate radius of the
Earth is 3,963 miles.]
2. How do the results of the algebraic and geometric approaches differ?
[The results are typically similar, but the algebraic
approach gives a formula that is an approximation, whereas the
geometric approach yields an exact value.]
3. Why is it important to make sure that the units of measure are the
same for the radius of the Earth and the height above sea level?
[The radius of the Earth is usually expressed in miles
or kilometers, whereas the height above sea level is usually expressed
in feet or meters. To use the Pythagorean theorem to find the
hypotenuse of the triangle formed by the height and radius, all
measurements must be converted to the same units.]
- Did the activities in this lesson provide sufficient challenges for all students? If not, how could the activity be modified?
- The activity sheet provides a fair level of guidance for
students. How could this lesson be presented (perhaps without the
activity sheet) to require more of students?
- Were students able to effectively compare the algebraic and
geometric solutions? What differences did they find when comparing the
- This lesson contains several activities for students. Rate the activities in this lesson from least to most challenging.
- How could this lesson be modified so that it could be taught without the activity sheet?
By the end of this lesson, students will:
- Generate a formula that can be used to predict the distance to the horizon from a given height above sea level.
- Determine the distance to the horizon from the summit of Mount Everest (29,035 feet).
Common Core State Standards – Practice
Make sense of problems and persevere in solving them.