Exploring Perimeter

If I had to suggest one theme for teaching students who historically have difficulty in mathematics class, it would be to make learning meaningful and purposeful. The following geometry problem has a few great qualities that make it perfect in accomplishing this goal. The solution can be derived from a combination of geometric skills, algebraic skills and/or simple arithmetic. This provides a number of access points for students. There is also a part of this problem that defies explanation, and that’s the hook.

Thanks to James Tanton for bringing this problem to my attention.

Imagine having a magic rope that wraps itself around the Earth at the equator. Suspend belief during this time, and also imagine that it falls perfectly on the Earth’s surface. Mountains or oceans do not affect it. The rope forms a perfect circle. You measure the distance of this rope, which provides the circumference of the Earth, then add 10 feet to it. Your new rope is now wrapped around the Earth, and since it’s a magic rope, it also hovers evenly over the surface of the Earth. What is the distance from the ground to the rope?

To tackle this problem, most students (and teachers), begin with the radius of the Earth and then calculate the circumference. From here a student can add the necessary 10 feet, work backwards until they have derived the new radius. Comparing radii offers a solution. It looks like this.

There is absolutely nothing wrong with this methodology, but what it fails to do is unlock the beauty of this problem. What if instead of the Earth, we were adding 10 feet of rope to the circumference of your coffee cup? What if we added 10 feet to the circumference of a penny? Would you believe me if the difference never changed?

On the left side of the equation I add 10 feet to the circumference, on the right I add an unknown quantity to the radius, which represents the distance between the radii. In the second step I sub in the definition of circumference for C (C=2πr). This allows me to solve for the unknown distance between the radii, exactly 1.59154!

With this knowledge in hand, invite your students to find circular objects, discover the circumference, add 10 feet to calculate the new circle and use string to trace the new circles. Students can now measure the differences between radii and experience the beauty of mathematics.

One last thing, do you really believe the magic rope would be 1.5 feet off the ground? Mindblowing.