How Far to the Horizon?

In geometry, the focus is on circles. We have finished a unit consisting of parts of circles, arc length, angles in a circle and segments in a circle. I decided to give them the following problem as a review:

10.1 How Far Is It To The Horizon Arc Length_1

I’ve been to Chicago a few times, and have gone up to the observation deck in the John Hancock building each time.

10.1 How Far Is It To The Horizon Arc Length_2

I originally played with this question in Algebra II and used this to set up a need for solving square root equations (which google was able to give me for the distance to the horizon…and a formula which one of my students found quickly on their phone). This time, in Geometry, I wanted to calculate the distance without any formulas. My students’ first question was how far off of the ground was that picture taken:

10.1 How Far Is It To The Horizon Arc Length_3

We discussed which of those heights was most appropriate and decided to use the height of the observation deck.

This is where I had to help them out a bit. I had to give my students a nudge in the right direction.

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They figured they needed to use circles, so they asked for the radius of the Earth.

10.1 How Far Is It To The Horizon Arc Length_4

As we drew on the Earth, my students noticed that my line of sight from the observation deck created a tangent line. We were able to create a right triangle, and use what we know about right triangle trigonometry to calculate the central angle created between the building and my line of sight. Once we had the central angle, we used arc length and calculated the distance to be approximately 39.3 miles.

10.1 How Far Is It To The Horizon Arc Length_5

And here was the answer based on what I was able to find online:

10.1 How Far Is It To The Horizon Arc Length_6

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