This was inspired by the flagpole guywire optimization post along with some recent events.

I recently had the need to drop a tree in some fairly tight quarters, where it was important that we know how tall the tree was to within a few feet.  There's a well-known problem in high school trigonometry that is very similar, where you use an angle measurement and a ground distance to compute the height of the tree.  

What we realized is that we could use the built-in spirit level in a cell phone to take the angle measurement.  By sighting along the edge of the phone to the top of the tree, we could get a fairly accurate angle measurement from the phone.




Basically, you can set the problem up as follows:



Where Theta is the measurement from the phone, X is a ground distance you can measure with a tape, and H is the unknown height of the tree.  Assume that the phone is up at 6 feet high when you're taking the measurement.

The math for calculating the height H of the tree when you have measured Theta and X is pretty straightforward trigonometry.  However, what we realized is that small errors in measuring Theta can have a big impact on the accuracy of your H estimate.  After some math, we realized that there was a way to minimize the effect of these errors by moving away from the tree such that we measured a specific angle Theta from the phone.

Question:
Assume that you can move forward and back as needed in order to get a particular measured angle Theta to the top of the tree.  What is the value of this angle that will maximize the precision of your calculation of H?

45 degrees.

Distance from tree to intersection of 45 degree line of sight to top of tree with ground equals height of tree.

(assumes level ground)

Back up until 45 degree line of sight intersects top of tree, measure distance from that point to center of tree, add height of your eye above ground to get height of tree.

45 degrees is spot on

Care to share your reasoning?

Geometry

Right triangles.

A 45, 45, 90 triangle has legs of equal length.

Since you are using your phone, take a look at the app “Theodolite “.

It’s got that feature built in.  

Measure from the tree base to where you are standing, put the crosshairs on the top of the tree, input the height of the phone, and calculate.

It is always better to understand the principle of what you are doing, rather than just poke buttons on a phone.

My apologies - I didn't realize I was in the Math and Science forum.  (I typically look at Active Topics and click on any thread that's interesting, without looking at which area forum I'm in. )

I'm not gonna argue that at all.  Understanding how it works is always better.

You can do the same thing with a stick, and if you never dropped a tree there is a lot more to it than knowing the height.

I'll be damned.  I didn't know that.  Thank you, as I still have a whole lot more trees to drop!

Use what the pros use:  Suunto clinometer (or for the tech-minded folks out there ) a laser hypsometer.

I use a PM5/66PC, with percent and topo scales.  Good for trees up to about 150 feet.  Only downside with a clino is you have to pull tape for your baseline; it's CRITICAL that your baseline is at the correct distance.

Attached File



The avoid that, a few years ago I got an affordable laser hypsometer, the Nikon Forestry Pro

YK1614 Nikon Laser Hypsometer by FredMan, on Flickr

The good thing about the Nikon is that it laser-ranges the distance to the tree, so you don't have to pull tape, and you can shoot multiple trees from the same location.  The bad thing is you HAVE to have a clear, unobstructed view horizontally to the tree.  And forget about using it in rain or fog, while it's waterproof the laser gets scattered by raindrops and fog and is useless.

It's ENTIRELY dependent on the accuracy of the meter in the phone, and how that accuracy changes as the actual angle changes.

The meter may be dead on at one range of angles and inaccurate at others, which is why, if you need truly accurate numbers, you use an instrument designed for the purpose.

Because at 45 degrees your X and your H are identical.  It's a right triangle with equal leg lengths.

If we had the money for that, we wouldn't be using a phone .

However, your first statement is not accurate.  I've worked extensively with the 3-axis digital MEMS accelerometers that are used in these devices- they are quite linear over their output range, and call be fairly well-calibrated for case alignment, bias, and sensitivity.

Your thought process is right on.    But, the accuracy of the iphone app is not what I would be comfortable with.    At least mine is not.

I admittedly forgot about this thread for a while until I saw somebody resurrected it.  Some of you have the correct answer, but have not provided a rigorous basis to say WHY it's correct.  

This problem is really a sensitivity analysis- what nominal angle do we want to measure so that we minimize the sensitivity of our height measurement to errors in measurement of the angle.  
This will involve some simple calculus- the harder part is figuring out how best to non-dimensionalize the problem.

It is easiest to work with the height of the tree minus the height of the user (6 feet).  We can estimate this reduced tree height \bar{H} as:


Our measured height \bar{H}_m involves an error H_error due to an error term epsilon in our measurement of the angle theta.  Substituting x from the above equation gives:


Solving for the height error and normalizing by the actual height \bar{H} will give us a fractional height error E_f:


We can find the change in fractional height error for a change in epsilon by taking a partial derivative, and evaluating at epsilon=0 (this will hold for reasonably small angular errors, on the order of a few degrees):


We could again differentiate this to find that it has a minimum (over the 0 to 90 degree interval) at 45 degrees, but it is easier to do via inspection:

This has a clear minimum sensitivity at 45 degrees- a bit over 3.5% of measured height per degree of measurement error.  As you get closer or further from the tree, your errors due to angular measurement error increase dramatically.

As a test, we can explicitly determine fractional error for a given scenario- 70 foot tree, 6 foot user, and a 45 degree nominal measurement angle.  A 1 degree error in measured angle corresponds to about a 3% error in measured tree height.

Bumping this

This is neat. Thanks for sharing it.