So, how high IS that antenna?

Recently I found myself once again shooting arrows over a tree for my XYL, (Dee, K5XYL), in order to hang another antenna for her amongst the tall Southern Pines surrounding our home. Honestly, I think she keeps me around JUST because I can string copper around to support her CW habit. I had decided to hang another "Carolina Windom" up for her and had found two perfectly spaced trees to act as the end supports for a flat-top configuration. After the requisite tangles, snapped string, disappearing arrows, etc..,. I had her antenna up and working. She then asked me "How high is it?" Now, I knew that was a loaded question, since if it was lower than the previous "Carolina Windom" I had put up for her, I would get the "I should have married someone with a two-hundred foot tower and 40 meter Yagi" look. I guesstimated about 70 feet and that got me off the hook, since the prior installation was at about 60 feet (also guesstimated, but don't tell her that!) Quickly changing the subject, I distracted her with a question about band conditions and made a mental note to myself (well, it was a mental note, who else would it be for???), to get an accurate height measurement.

Somewhere in the back of my mind, a hazy memory from trigonometry back in high school lurked and resurfaced during my time of need….something about an inclinometer. A few more seldom accessed neural pathways fired and then I remembered. I have seen several methods for determining the height of an object. This one is probably no better, worse or accurate than most, and if you are an engineer, have done survey work, work for a forestry agency or grew up participating in model rocketry, you will probably laugh at how simple and obvious this is, but for everyone else, it will at least allow usage of some of those math skills that you swore to yourself you would never use again once you finished school! It also involves "Super Glue", a protractor, string and a tape measure so it can help bring out your inner child as well (it is also a cool project to enjoy with a youngster if you happen to have one hanging around. Borrow one if you need to. Making science fun is a great way to help Elmer the next generation of hams.)

What you'll need:

Half circle plastic protractor with a small hole at the origin. (Standard, not decimal or fractional)

One straight soda straw

Super Glue (or tape)

One foot of thin twine or string.

A small weight (1/4 ounce fishing sinker or 1/4 inch nut or similar)

Tape measure (any length will do, but a 25 footer makes it quicker)

Calculator (The built in Scientific calculator for the Windows XP OS will be fine)

Construction details: (or how to make an inclinometer for the ultra-cheapskate)

Most of these cheap protractors are about six inches long at the base, so cut your straw about seven inches long. Prop your protractor up between two books (the protractor will be perpendicular to your work surface) with the curved surface against (and tangent to... ha more math stuff, snuck that one in..) the work surface. This will leave the straight edge pointed upwards and parallel to your work surface (ummm... DON'T prop it up with anything that you can't afford to lose if you squirt "Super Glue" all over the place by accident! I used some of my wife's 'Chick Lit' books, hi hi!). Run a bead of glue across the thin edge (usually about 1/16 inch) of the flat, upward facing surface. Quickly put your soda straw onto this glue surface, with the straw extending past one end of the protractor about an inch or so. This will make it easier to hold up close to your eye. Once the glue has set and you have unglued your hands from the table or books or some part of your anatomy, and unglued any youngster who has been 'helping' you, you are ready to add the business part of the inclinometer, which is a string with a weight tied onto the end. Take your string and run it through the hole at the origin of the protractor (that's the hole right in the center of the base that will be directly below the 90 degree mark). Tie a big wad of knots in the string on the backside of the protractor to keep the string from pulling through the hole. Pull the string through the front side of the protractor until the knots snug up against the hole. Now tie your weight on the other end. The length of the string hanging down doesn't really matter, as long as it hangs past the edge of the protractor angle markings. Six to ten inches should be fine. That is it. You are done. Go grab an adult beverage for yourself and a soda for any youngster you had helping, assuming you didn't 'accidentally' glue their mouth shut after asking you how much longer this was going to take for the two-hundredth time.

So, now that you are the owner of a fine, scientific instrument, how do you use it? You could just leave it lying around and mumble things about 'apogee, gradients, look angle' etc... when someone asks what the heck it is, then nod sagely and walk off. OR you can take it outside and find out if your dipole is really at 100 feet (probably not!).

I promise that the math involved here has never, to my knowledge, killed anyone. If you are worried about that, have your mother-in-law do the math for you (sorry Kathy!).

From the base of the support structure for you antenna (i.e. tree), use your tape measure to measure a spot between 50 and 100 feet away. You can actually use ANY distance, as long as it is known, but 50 -100 feet is usually far enough away for you to use the inclinometer without the weight smacking you in the chin. You also need a spot that is at least roughly level with the base of your tree. If you are significantly higher or lower than the base of your tree, you will have to take that into account. Once you are at your known distance, hold the inclinometer up to your eye, with the straw on top, the protractor perpendicular to the ground you are standing on, and sight through the straw to where your support rope is hung. If your support rope is way, way, way out on a limb, you'll have to estimate a spot on the ground directly underneath where your rope crosses the limb, then measure your 50 to 100 feet from there to get an accurate result. Hopefully you still have one of those youngsters around, and get them to see where the string intersects the degree markings on the protractor. Most protractors will have two sets of numbers on them, so you want the number that is less than 90-degrees. If your protractor is REALLY cheap, I suppose that it might only have one set of degree markings on it. In that case, make sure the end that starts at 0 is the end closest to your eye. Anyway, if your measured angle is 90-degrees or higher, you are doing something wrong or the laws of the natural world have been suspended and you might as well go have another adult beverage. If you have no youngster around, just use your free hand to hold the string at the free hanging angle, then get your measurement. So now you have an angle and a known length. We are going to assume that your support structure (tree) is relatively straight and perpendicular to the ground and that you were sighting on a spot close to the trunk. This means that we have a right triangle, with one leg formed by your distance from the tree, one leg formed by the distance up the tree to your antenna support and all of the angles of the triangle (one angle is 90-degrees, one angle is what you measured (the angle at the top of our triangle), subtract the sum of those from 180-degrees and you know the angle at the bottom! For example, we will assume that I was standing 100 feet from the base of the tree, and I measured a 55-degree angle sighting on where the limb that my antenna support rope is hung is attached to the tree. I now know that one leg of my triangle is 100 feet, the angle at the top of the triangle is 55-degrees, the angle at the base of the triangle is 90-degrees and the remaining angle is 35-degrees (180 - 55 - 90 = 35). We can now use very basic right angle trigonometry to solve for my unknown triangle leg (i.e. the height we are looking for). Here is how:

The cosine of an angle = adjacent side length / hypotenuse length. This is normally shortened to: Cosine equals adjacent over hypotenuse. Here is how to plug in the numbers in our example:

cos 35 degrees = 100 over (or divided by) the unknown length of the hypotenuse which is the distance between where we were standing when we took the measurement and the point we were looking at through the straw. Moving the numbers around, we come up with:

length of the hypotenuse equals 100 divided by the cosine of 35-degrees. On your calculator you would type:

100 / 35 cos =

This would give you about 122 (rounded) for the length of the hypotenuse. Now we know the length of one leg and the length of the hypotenuse of a right triangle. This means we can use the old Pythagorean theorem (C squared = A squared + B squared) to find the height we are looking for. In this case it is: 122 squared = 100 squared + B squared. Solving for B squared (unknown height squared) we get:

B squared = C squared - A squared

height squared = 122 squared - 100 squared

height squared = 14844 - 10000

height squared = 4844

height = square root of 4844

height = 70 feet (rounded)

That is it..... well, almost. Since you are probably 5 to 6 feet tall, you were actually taking your angle measurements about 5 feet above the ground. Add this height to your result and you should be good to go, or get really close to the ground when you take your angle measurement and forget the slight error. If you wanted to be really technical, you could take the distance from your eyes to the ground, and the same angle that you measured at the top of your triangle and compute where the hypotenuse would actually terminate at the adjacent leg (if it were at ground level and extended far enough out). Do this by taking the tangent of the angle formed by your body and the now extended hypotenuse (which is the same angle as you measured with your inclinometer) multiplied times the distance of your eyes to the ground and add this number to the adjacent leg length. This would give you a new length for the adjacent leg (now at ground level and about 107 feet in my example with a 5 foot eye-to-ground distance) Then plug all of this back into the original equation like this:

107 / 35 coos =

Plug this number (131) into the next equation:

Height squared = 131 squared - 107 squared and you end up with:

75 feet...

Whew...

If you really are math averse (bless you!), you can avoid the math like this:

Take your inclinometer and sight in on your target. Take a reading. If it is less than 45 degrees, move away from your tree and continue to take readings until you get to a 45-degree angle. Do the opposite if your initial reading is greater than 45 degrees. Once you get a 45-degree angle, mark the spot on the ground. Now just use your tape measure and measure from that spot to your tree. You now have an isosceles triangle, and the distance from your mark to your tree will also be the height you are looking for. Of course, you'll still need to correct for the eye to ground distance. That was MUCH easier, huh? Not nearly as much fun, though.

I know, I know..... If you are just putting your antenna up, you could mark your rope every five feet or so with some tape then count the marks as you were putting it up. OR, if you are using thin twine to pull up your heavier support rope, just pull the rope up to the highest point, and mark it at the bottom then pull it down and measure it with a tape measure. Or..... Well, there is a bunch of ways to do this. I like this one because I get to exercise the old noggin a little bit. You could use this technique to figure out if that tree that is leaning towards your house is tall enough to be a worry. Or how high the center of your flat top is. Or how high the peak is of a roofline that you want to tack on a j-pole to. The inclinometer can be used for a lot of other things too. Quick and dirty way to figure out your look angle if you are trying to hit a satellite or measuring the steepness of a slope.

I apologize to all of you math heads out there for putting you through the mangled high school trig, but for the rest of us, I hope this is helpful. When someone asks you how high your dipole is, you can now lie with authority and say 100 feet!

Ted Forrest

KB5CQF

(No mother in laws were hurt during the writing of this article)