My last post explains how I introduce radian angle measure to my students. That post generated three interesting questions about wrapping a piece of string around a great circle of the Earth. The first two questions (see below) are familiar to many math teachers and students of mathematics. However, I have never seen a question similar to question 3, nor was I was able to find a question similar to it discussed on the web despite abundant resources that provide solutions to math questions. Because the third question is new and interesting to me, and it relates to the content of my last post, this post focuses on solving the third question. If you would like to review detailed solutions for the first two questions, open this PDF file.
The solutions of the three questions can be explained in terms of the radius r of a circle, the length L of an arc that subtends central angle θ, and the radian measure of θ. The relationships L/r = θ and L = rθ were used to derive the equations D = 2πd and d = D/(2π). D equals the difference between the circumference of a great circle ring around a sphere and the circumference of a sphere, and d equals the uniform distance between the great circle ring around the sphere and the surface of the sphere. Because the equations D = 2πd and d = D/(2π) apply to any sphere, not just planet Earth, results derived from these equations are interesting and counter intuitive.
Question 1: Imagine that the surface of planet Earth is like the surface of a perfect sphere. Consider a piece of string, longer than the circumference of the Earth, that is used to form a great circle ring around the Earth. If the uniform distance d between the outer ring and the surface of the Earth equals 1 inch, how much longer than the circumference of the Earth must the circumference of the circular ring be?
Solution: D = 2πd = 2π(1 inch) = 6.28 inches
Question 2: Imagine that the surface of planet Earth is like the surface of a perfect sphere. Consider a piece of string, 1 inch longer than the circumference of the Earth, that is used to form a great circle ring around the Earth. What is the uniform distance d between the outer ring and the surface of the Earth?
Solution: d = D/(2π) = 1 inch /(2π) = 0.1592 inches
Question 3: Imagine that the surface of planet Earth is like the surface of a perfect sphere. Consider a loop of string, 1 inch longer than the circumference of the Earth, that is wrapped along a great circle of the Earth and pulled at a single point A so that the loop of string is taut. What is the distance between point A and the surface of the Earth? The answer to this question, 413.2237 inches, is very surprising.
The diagram below shows the setup for solving question 3. From Euclidean geometry, we learn that a radius to a point of tangency is perpendicular to the tangent line.
The text box below gives the key steps in solving question 3. Of course, readers will find that I skipped some steps when going from one step to the next. A suggestion for the dissatisfied or skeptical reader is to find a quiet corner of his/her dwelling, and write out the complete solution by making the appropriate substitutions and simplifying the resulting equations. The method that I used to solve the trigonometric equation is not found in typical math text books. If you find an error in my solution of question 3, or find a more elegant solution, please reply to this post, or send an email to email@example.com. I’m still amazed that d ≈ 413. 2237 inches!
A Suggestion for Teachers
After a classroom discussion of question 3, trig and pre-calculus teachers have a golden opportunity to transform their classroom into a mathematics laboratory for a day.
- Organize the class so that each student has a lab partner.
- Provide each lab partnership with a toy model of a sphere, a piece of string, and adhesive tape.
- Provide the appropriate tools to measure the diameter of the sphere and string length.
- Construct a string-sphere model similar to the model described in question 3.
- Measure the distance from point A to the surface of the sphere.
- Calculate the expected distance from point A to the surface of the sphere.
- Create a written report that includes a discussion of their model, experimental measures, and a complete description of how the expected distance was calculated.
- The written report should also include a discussion of the experimental error found in their results.