Clinometers: Measuring Height of Objects with Isosceles Right Triangles (May 27 and 28 only)

Although I could continue with binary lessons for months, I wanted to end the season with an exercise that would teach children a real life skill when there are no computers available. 

All year they have been earning 100 dollar Mathlete notes with Pythagorus of Samos front and center. On this note, there is a picture of a right triangle with an example of the Pythagorean Theorem. While studying right triangles in the fourth century BC, his students, called the Pythagoreans, discovered amazing properties of the isosceles right triangle: the ratio of the sides was always 1:1:root 2 (or about 1.41). The children were given a single square sticky note and told to create an isosceles right triangle with just folding. Some did it with one fold, some with repeated folds. We came up with 6 different sizes but all similar triangles. 

Mariners invented the clinometer (or astrolabes) to help with finding latitudes and longitudes at sea. Today’s uses of clinometers is in forestry (finding heights of tree’s or percent grade on a slope), survival at sea, and wilderness survival. Clinometers are small enough to fit into a backpack should your GPS (global positioning system fail). These survival techniques include finding latitude from the Pole Star Polaris at night or latitude and longitude from the mid-day sun.

We chose to study clinometers as they relate to finding heights of objects as long as you can measure the distance from your position to the object. The right isosceles triangle has two legs that have identical length. Look at the diagram on the attached pdf to see how the right triangle is positioned against the tree. If you find the top of an object using the clinometer with a reading of 45 degrees (half of your 90 degree angle), your horizontal distance to the object is almost exactly the same as the height of the object. I say almost exactly because you have to add one more measurement, the distance from the ground to your eye.

The children made their own clinometers and should take them everywhere until the end of school and during the summer. They used the Clinometer Template (attached pdf) and cut it out of card stock. They had to punch a hole in the vertex of the right angle and put a string through it and tape it to the back. They tied two paper clips to the other end to dangle with gravity at 45 degrees. Finally, they taped a straw to the 0 degree line. I measured their individual distance from ground to their eye which they will have to add to every calculation.

If you see a building or a tree or a pole, measure its height and record it. For example, we measured one of the Mathlete host houses to be exactly 28 feet. One of the children who has a distance from ground to eye of four feet, used the clinometer to put herself at a distance from the house of exactly 24 feet. How did she measure the 24 feet. First I had all of the children practice a step of 2 feet along measuring tapes. This is the average step of an elementary school child. However, everyones gait is different (as unique as fingerprints), so this is something to practice. She took 12 paces to the house and then multiplied by two. We then used the tape measure to verify the calculation and it was exact. 

My natural step is exactly 3 feet so I took exactly 8 paces to the house. Here is where it gets tricky, if your natural gait is 1.75 feet or 2.25 feet, the multiplication is more difficult; although doable. Practicing to pace off 2 feet is much easier. When you get older, you will most likely be able to practice a three foot step.

If you are a curious mathematician like the Mathletes you will take your clinometer on vacation as I do. Last March, I calculated the height of the Eiffel Tower in Paris to within 5 feet (it is 986 feet). Last November, I calculated the height of the Empire State Building along 34th street in New York City to within 3 feet (to the tip it is 1454 feet). A word of caution: you need to step off all the way to the middle of the structure to get the height since the very apex (top) is vertically above the middle of the base.

I was unable to calculate the height of the Statue of Liberty since I could not stand far enough away from it at 45 degrees with the clinometer. I would have been in the water.

Note for the future: in high school, you will learn trigonometry which will allow you to use the clinometer using any angle between 0 and 90 degrees. For right now, we will rely on the simple 45-45-90 triangle.