How High Is That?

If you want to figure out how tall something is (like a house or tree), or how high it goes (like a rocket or balloon), there’s a simple and inexpensive way to get a fair estimate.

You’ll need a couple things for this, but they’re easy to find and you probably already have them around the house. Find a protractor and a rectangular piece of cardboard bigger than the protractor. You’ll also need a push pin, some string and a weight of some kind (I used a fishing sinker).

First, let’s make a simple theodolite, which is a tool used to measure vertical angles. Take the cardboard and using the push pin, fasten the protractor to it so that the flat edge of the protractor runs along the top of the cardboard. Tie the weight to one end of the string and the other end to the pin. This way, when you tilt the cardboard you can read the angle by seeing where the string hangs past the protractor.

The other thing you’ll need is a tangent table, which can be found in any trigonometry textbook. That’s right, you’re using trigonometry for this! Use the one below, or find one to your liking, they're all the same (click it and it gets bigger).

Still with me? Good! Believe me, this is simple. In fact, this explanation takes longer than the process. The figure below shows the basic concept of determining height or altitude:

Take the theodolite and stand a known distance from what you’re trying to measure. In the diagram, it's where the black and blue lines meet. This distance is the baseline, and the farther the better (as long as you can see the top of the thing you’re measuring). For instance, say you’re going to measure the altitude of a model rocket, and you’re launching from a football field. The tracker is on one goal line, 300 feet (100 yards) away from the launch pad on the other goal line. When the rocket launches, the tracker follows the rocket with the theodolite until the rocket reaches apogee (it's highest point). The angle is read (where the string marks it on the protractor), and this angle is written down.

Time for some simple math. The formula is on the diagram. Look up the tangent for the angle on the table, multiply that number by the baseline, and that is the altitude in feet. Simple!!!
An example: baseline is 300 feet and your measured angle is 40 degrees. The tangent for 40 degrees is .839, so 300 * .839 = 251.7 feet.

This technique works great for things that stand still or go straight up, but the measurement will be off if there’s any horizontal movement. Using our model rocket example again, if the rocket curves towards you on the way up, then your measured angle will increase and the calculated altitude will be too high. One way to compensate for this is to have two people with theodolites standing at 90 degrees from each other (imagine a rocket launching from home plate on a baseball diamond and trackers standing on first and third bases). You can average their measurements and get a pretty good estimate of the correct height.

You can also make a sturdier theodolite by replacing the cardboard with a length of wood or broomstick. Screw the protractor into the side of the wood, hold the theodolite like a rifle and sight along the length of it to get your measurement. You can drive a couple of finishing nails into the top of the wood to help with your sighting if you want, but it's not strictly necessary.

So, how high is the tallest tree in your neighborhood?