## Cosine method

### Cosine method

You see a tree and all you have is your laser rangefinder. You have no clinometer, theodolite or transit with you. Can you still measure the height without knowing any angle? Yes. Use the cosine method.

If the ground is flat and the top is directly above the base, use the laser to measure the distance to the top, and the distance to the base. Divide the second measurement by the first and you get a fraction. On the calculator, hit inverse and cosine and you get the angle you would have gotten with the clinometer. Hit sine and then multiply by the distance to the top, you get the height of the tree.

If the ground is flat and the top is not directly above the base, have somebody stand directly under the top, or place an object there that the laser can measure to. Or, go to a spot that is equidistant from the base and the spot under the top. Do the same method as above.

If the ground is not flat and the base is below you, and the top is directly above the base, measure to an arbitrary point on the tree as close to the horizontal as you can, and measure to the top. Divide the first measurement by the second and do an arc-cosine, you get the upper part of the height. Then measure to the base and divide the horizontal measurement from before by this measurement, do an arc-cosine and you get the lower part of the height, add that to the upper part of the height from before.

If the ground is not flat and the base is below you, and the top is not directly above the base, have somebody stand under the top with his hand up or holding a pole so you can get a measurement on the horizontal. Divide that by the measurement to the top and do an arc cosine. Divide that same horizontal measurement by the measurement to the the guy's feet and do an arc cosine. Add the two.

If the ground is not flat and the base is above you, you can't use the cosine method.

If the ground is flat and the top is directly above the base, use the laser to measure the distance to the top, and the distance to the base. Divide the second measurement by the first and you get a fraction. On the calculator, hit inverse and cosine and you get the angle you would have gotten with the clinometer. Hit sine and then multiply by the distance to the top, you get the height of the tree.

If the ground is flat and the top is not directly above the base, have somebody stand directly under the top, or place an object there that the laser can measure to. Or, go to a spot that is equidistant from the base and the spot under the top. Do the same method as above.

If the ground is not flat and the base is below you, and the top is directly above the base, measure to an arbitrary point on the tree as close to the horizontal as you can, and measure to the top. Divide the first measurement by the second and do an arc-cosine, you get the upper part of the height. Then measure to the base and divide the horizontal measurement from before by this measurement, do an arc-cosine and you get the lower part of the height, add that to the upper part of the height from before.

If the ground is not flat and the base is below you, and the top is not directly above the base, have somebody stand under the top with his hand up or holding a pole so you can get a measurement on the horizontal. Divide that by the measurement to the top and do an arc cosine. Divide that same horizontal measurement by the measurement to the the guy's feet and do an arc cosine. Add the two.

If the ground is not flat and the base is above you, you can't use the cosine method.

Last edited by morgan on Tue Oct 01, 2013 10:09 pm, edited 1 time in total.

### Re: Cosine method

Morgan,

No, don't use the cosine method. That is the biggest problem with the bad measurements that are out there. It would work if everything is in an ideal situation as you said, but the top of the tree is rarely directly over the base. People think that the top is over the base when it is not. They think they can identify which spike is the top of the tree when they can't. The wrong top is identified as often as not even by more experienced measurers. The error from the top being offset is equal to the tangent of the angle to the top times the amount of offset. Yes if the situation is ideal you can get a good number, but you can't really tell if the situation is ideal or not in most cases, so you can't tell if your numbers are good or not. Tree heights measured with the cosine method are not acceptable as an NTS measurement and not acceptable for inclusion in the database. The point to remember is that while you can get good numbers, there is no way to consistently tell if the numbers for a particular tree are good or not, so NONE of them can be accepted as a good measurement.

In this situation your best bet to get get a height, and get one that is acceptable, is to shoot straight up from somewhere near the trunk hitting as high of a branch as you can see. If the angle to the top is 80 degrees or more the difference between straight up and this angle at most 1.5%. The tree height will not be exaggerated (or exaggerated by less than 1.5% if you hit the very tip of the highest branch at 80 degrees) and likely will underestimate the height of the tree by some amount because of failure to hit the extreme top of the branch. Anything that can result in the tree height being exaggerated by more than a nominal amount simply can't be accepted if we want to keep our data set clean. Under estimates really don't distort the data set any more than simply not measuring it at all would do. A better measurement can always be taken at a later time. If someone wants to measure the tree height and get an accurate measurement, they should get both a laser rangefinder and a clinometer, especially as the latter are available as a free or $1 app on many smart phones if you can't afford a real clinometer.

Bob Leverett has proposed many alternative ways to measure tree height based upon using just a clinometer, or with this or that equipment. These are fine as a mathematical exercise and are valid. Your comments about the cosine is workable (although I think the distance to the tree needs to be horizontal for the process to work), but the practical side of me says The measurements would be easier and much more accurate if the measurer simply used both a clinometer and laser rangefinder.

Edward Frank

No, don't use the cosine method. That is the biggest problem with the bad measurements that are out there. It would work if everything is in an ideal situation as you said, but the top of the tree is rarely directly over the base. People think that the top is over the base when it is not. They think they can identify which spike is the top of the tree when they can't. The wrong top is identified as often as not even by more experienced measurers. The error from the top being offset is equal to the tangent of the angle to the top times the amount of offset. Yes if the situation is ideal you can get a good number, but you can't really tell if the situation is ideal or not in most cases, so you can't tell if your numbers are good or not. Tree heights measured with the cosine method are not acceptable as an NTS measurement and not acceptable for inclusion in the database. The point to remember is that while you can get good numbers, there is no way to consistently tell if the numbers for a particular tree are good or not, so NONE of them can be accepted as a good measurement.

In this situation your best bet to get get a height, and get one that is acceptable, is to shoot straight up from somewhere near the trunk hitting as high of a branch as you can see. If the angle to the top is 80 degrees or more the difference between straight up and this angle at most 1.5%. The tree height will not be exaggerated (or exaggerated by less than 1.5% if you hit the very tip of the highest branch at 80 degrees) and likely will underestimate the height of the tree by some amount because of failure to hit the extreme top of the branch. Anything that can result in the tree height being exaggerated by more than a nominal amount simply can't be accepted if we want to keep our data set clean. Under estimates really don't distort the data set any more than simply not measuring it at all would do. A better measurement can always be taken at a later time. If someone wants to measure the tree height and get an accurate measurement, they should get both a laser rangefinder and a clinometer, especially as the latter are available as a free or $1 app on many smart phones if you can't afford a real clinometer.

Bob Leverett has proposed many alternative ways to measure tree height based upon using just a clinometer, or with this or that equipment. These are fine as a mathematical exercise and are valid. Your comments about the cosine is workable (although I think the distance to the tree needs to be horizontal for the process to work), but the practical side of me says The measurements would be easier and much more accurate if the measurer simply used both a clinometer and laser rangefinder.

Edward Frank

"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

### Re: Cosine method

It's impossible for the cosine method to overestimate the height. It doesn't use the angle to a false top the way the tangent method does. It uses an actual laser generated hypotenuse which can never be greater than the true hypotenuse would be. Let me explain:

1. If the competing top you are measuring is closer to you and has a greater angle than the true top, so it LOOKS like a higher top, the tangent method would give a false reading to an imaginary top which isn't there. The cosine method, on the other hand, since it measures distance and not angle, would give a shorter distance than the real top, as if measuring to a lower level on the tree. It would therefore underestimate the height.

2. If the competing top you are measuring is on the far side of the tree and farther than the true top, it would be blocked by the true top, and even if it weren't, it would never look taller than the true top so you'd never be measuring it in the first place.

3. If the competing top and true top are equidistant from you, the competing top would give a lower reading than the true top.

1. If the competing top you are measuring is closer to you and has a greater angle than the true top, so it LOOKS like a higher top, the tangent method would give a false reading to an imaginary top which isn't there. The cosine method, on the other hand, since it measures distance and not angle, would give a shorter distance than the real top, as if measuring to a lower level on the tree. It would therefore underestimate the height.

2. If the competing top you are measuring is on the far side of the tree and farther than the true top, it would be blocked by the true top, and even if it weren't, it would never look taller than the true top so you'd never be measuring it in the first place.

3. If the competing top and true top are equidistant from you, the competing top would give a lower reading than the true top.

### Re: Cosine method

NTS,

I'm pretty weighted down with the upcoming Atlanta event, the American Forests initiative, and other projects, but I'll try to make some time to present a set of diagrams that show what works provided the assumptions behind the diagrams are met.

What would work as a height measurement if you have a laser rangefinder, but not a clinometer, assuming the ground is level and the tree's top is vertically positioned over the base? Let's look at the diagram.

If the baseline is level, or reasonably so, and the top is vertically over the base, then you can simply apply the Pathagorean Theorem to get height above eye level and then add your height. Since you have a laser rangefinder, you can measure the hypotenuse and adjacent side of the right triangle that goes from your eye level to the trunk and vertically up to the top and back to your eye. Alternatively, you can use the adjacent side and hypotenuse to calculate the angle to the top using the inverse or arc cosine function. However, all that does is give you an angle, not a height. But, you then take the sine of the angle times the hypotenuse to get the height above eye level and add your height. Or you could take the adjacent side times the tangent of the angle plus your height. I re-emphasize that in this first diagram that the ground is level and the top is positioned vertically over the base (and obviously visible). The strategy here is that although you don't have a clinometer to determine eye level on the trunk, you can go to the trunk and mark eye level on it and shoot from your vantage point to the marked point on the trunk.

I'll address a more complicated (and realistic) scenario tomorrow and what can and cannot be done in terms of height measurements. This stuff is not rocket science, but you do have to know the mathematics. As a minimum, to handle just about any thing that can be thrown at you, you need to understand:

1. Types of plane triangles, e.g., acute, right, obtuse, isosceles, equilateral

2. Laws that govern plane triangles, e.g. sum of angle equal 180 degrees.

3. Right triangle trigonometry (sine, cosine, tangent, arcsine, arccosine, arctangent)

4. Law of cosines

5. Law of sines

6. Pathagorean theorem

7. Principle of similar triangles

You also need to be able to algebraically solve for unknowns in equations that may not be set up as convenient formulas with a variable on the left side and knowns on the right. Many practical problems require algebraic manipulation, or some one who can set up the final formulas for you. Finally, any plane triangle can be solved if you know:

1. Two sides and an angle

2. One side and two angles

3. Three sides

If your triangle doesn't fit one of these three scenarios, it can't be solved in terms of determining all sides and angles. In the example in the diagram, the assumption is that the triangle is a right triangle. That means we know one angle. Since two sides were measured (adjacent and hypotenuse), we meet the conditions of scenario #1.

Bob

I'm pretty weighted down with the upcoming Atlanta event, the American Forests initiative, and other projects, but I'll try to make some time to present a set of diagrams that show what works provided the assumptions behind the diagrams are met.

What would work as a height measurement if you have a laser rangefinder, but not a clinometer, assuming the ground is level and the tree's top is vertically positioned over the base? Let's look at the diagram.

If the baseline is level, or reasonably so, and the top is vertically over the base, then you can simply apply the Pathagorean Theorem to get height above eye level and then add your height. Since you have a laser rangefinder, you can measure the hypotenuse and adjacent side of the right triangle that goes from your eye level to the trunk and vertically up to the top and back to your eye. Alternatively, you can use the adjacent side and hypotenuse to calculate the angle to the top using the inverse or arc cosine function. However, all that does is give you an angle, not a height. But, you then take the sine of the angle times the hypotenuse to get the height above eye level and add your height. Or you could take the adjacent side times the tangent of the angle plus your height. I re-emphasize that in this first diagram that the ground is level and the top is positioned vertically over the base (and obviously visible). The strategy here is that although you don't have a clinometer to determine eye level on the trunk, you can go to the trunk and mark eye level on it and shoot from your vantage point to the marked point on the trunk.

I'll address a more complicated (and realistic) scenario tomorrow and what can and cannot be done in terms of height measurements. This stuff is not rocket science, but you do have to know the mathematics. As a minimum, to handle just about any thing that can be thrown at you, you need to understand:

1. Types of plane triangles, e.g., acute, right, obtuse, isosceles, equilateral

2. Laws that govern plane triangles, e.g. sum of angle equal 180 degrees.

3. Right triangle trigonometry (sine, cosine, tangent, arcsine, arccosine, arctangent)

4. Law of cosines

5. Law of sines

6. Pathagorean theorem

7. Principle of similar triangles

You also need to be able to algebraically solve for unknowns in equations that may not be set up as convenient formulas with a variable on the left side and knowns on the right. Many practical problems require algebraic manipulation, or some one who can set up the final formulas for you. Finally, any plane triangle can be solved if you know:

1. Two sides and an angle

2. One side and two angles

3. Three sides

If your triangle doesn't fit one of these three scenarios, it can't be solved in terms of determining all sides and angles. In the example in the diagram, the assumption is that the triangle is a right triangle. That means we know one angle. Since two sides were measured (adjacent and hypotenuse), we meet the conditions of scenario #1.

Bob

Robert T. Leverett

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre

### Re: Cosine method

Morgan,

Ok. I understand what you are saying. Let's see what Bob comes up with to formalize it a bit more. If there aren't any problems found, congratulations on adding a new methodology!

Ed

Ok. I understand what you are saying. Let's see what Bob comes up with to formalize it a bit more. If there aren't any problems found, congratulations on adding a new methodology!

Ed

"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

### Re: Cosine method

Thanks Ed. It's basically the sine method that NTS favors, but if you have no clinometer you can use the arc-cosine to figure out the angle you would have gotten if you had a clinometer. Then do a sine with that angle. Or just the pythagorean theorum.

Like Bob said, not rocket science, but it's good that you can measure a tree accurately with just a laser.

The reason the cosine method fails if the base of the tree is above you, as I mentioned in my original post, is, you can't measure the horizontal length to the tree without digging a ditch or getting on a stepladder. The triangle you would get from top to base to eye would not be a right triangle, and trigonometry doesn't work if it's not a right triangle. Neither does the pythagorean theorum. But you can walk around to the other side of the tree and measure from there!

Like Bob said, not rocket science, but it's good that you can measure a tree accurately with just a laser.

The reason the cosine method fails if the base of the tree is above you, as I mentioned in my original post, is, you can't measure the horizontal length to the tree without digging a ditch or getting on a stepladder. The triangle you would get from top to base to eye would not be a right triangle, and trigonometry doesn't work if it's not a right triangle. Neither does the pythagorean theorum. But you can walk around to the other side of the tree and measure from there!

### Re: Cosine method

Morgan, Ed, et. al.

Here is the next step. We'll keep the assumption that the tree is vertical, but that the ground is not. So, we have slope to contend with. There is a solution. The diagram shows one way to solve the problem. We look for the shortest distance from eye to the trunk from our vantage point. So, we would be scanning up and down the trunk until we settle on the spot on the trunk that is closest. Bear in mind, we hold on to the assumption that the top is vertical over the base and that the trunk is also straight and vertical.

I realize that these laser only methods are not ideal, but if we have only the one instrument, what can we do to measure the tree using other tricks of the trade? This is really what Morgan is getting at, and it opens up a whole new line of investigation. There are plenty of geometrical constructions that we could employ that would allow us to solve a problem with minimum instruments. We can go along way in this direction. Let's face it, ancient civilizations did a lot of this sort of measuring. And the mathematical constructions are in most basic plane geometry books.

When it comes to mathematics, people often move away from it as rapidly as they can when they leave high school or even college. That branch of science is then left to those with special aptitudes who create wizardry beyond the interests and capabilities of ordinary folks. However, Morgan has pointed to basically a new direction. We can apply ourselves and develop novel, minimal equipment-based solutions using fairly basic mathematics.

What we have to do is be willing to recognize the assumptions behind each approach. The one size fits all approach because it is convenient or is tradition has no place in NTS. Viva la innovation. More stuff coming as I can squeeze out some time. Oh yes, because the displays for most laser rangefinders only read to the yard or half yard, one needs to apply the point of change-over technique when getting the distance to the point that is presumed to be close to level.

BTW, double click on the image to expand it for better visibility.

Bob

Here is the next step. We'll keep the assumption that the tree is vertical, but that the ground is not. So, we have slope to contend with. There is a solution. The diagram shows one way to solve the problem. We look for the shortest distance from eye to the trunk from our vantage point. So, we would be scanning up and down the trunk until we settle on the spot on the trunk that is closest. Bear in mind, we hold on to the assumption that the top is vertical over the base and that the trunk is also straight and vertical.

I realize that these laser only methods are not ideal, but if we have only the one instrument, what can we do to measure the tree using other tricks of the trade? This is really what Morgan is getting at, and it opens up a whole new line of investigation. There are plenty of geometrical constructions that we could employ that would allow us to solve a problem with minimum instruments. We can go along way in this direction. Let's face it, ancient civilizations did a lot of this sort of measuring. And the mathematical constructions are in most basic plane geometry books.

When it comes to mathematics, people often move away from it as rapidly as they can when they leave high school or even college. That branch of science is then left to those with special aptitudes who create wizardry beyond the interests and capabilities of ordinary folks. However, Morgan has pointed to basically a new direction. We can apply ourselves and develop novel, minimal equipment-based solutions using fairly basic mathematics.

What we have to do is be willing to recognize the assumptions behind each approach. The one size fits all approach because it is convenient or is tradition has no place in NTS. Viva la innovation. More stuff coming as I can squeeze out some time. Oh yes, because the displays for most laser rangefinders only read to the yard or half yard, one needs to apply the point of change-over technique when getting the distance to the point that is presumed to be close to level.

BTW, double click on the image to expand it for better visibility.

Bob

Robert T. Leverett

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre

### Re: Cosine method

Bob, is it possible to erase the little leg of red on H below the horizontal? H should run from D to L, am I correct?

Where did you get the little green forester with the rangefinder, it's awesome.

Where did you get the little green forester with the rangefinder, it's awesome.

### Re: Cosine method

Morgan,

You are absolutely right. I forgot to stop the vertical red line at the end of D. That is where the vertical red line should end. The diagram below corrects the problem. Oh yes, and the funny little guy is from Microsoft clip art under the people category. It's actually a detective with a magnifying glass, I think, but it works for our purposes.

Bob

You are absolutely right. I forgot to stop the vertical red line at the end of D. That is where the vertical red line should end. The diagram below corrects the problem. Oh yes, and the funny little guy is from Microsoft clip art under the people category. It's actually a detective with a magnifying glass, I think, but it works for our purposes.

Bob

Robert T. Leverett

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre

### Re: Cosine method

NTS,

In getting ready for the measuring classes (beginning, intermediate, and advanced) that Will Blozan and I will be presenting at the TCI-NTS rendezvous on Oct 10-12, I've been assembling materials. I expect that Images speak loudest. Any comments on the one below? Thanks in advance. Be sure to double click on the image to expand it so that it is readable.

Bob

In getting ready for the measuring classes (beginning, intermediate, and advanced) that Will Blozan and I will be presenting at the TCI-NTS rendezvous on Oct 10-12, I've been assembling materials. I expect that Images speak loudest. Any comments on the one below? Thanks in advance. Be sure to double click on the image to expand it so that it is readable.

Bob

Co-founder, Native Native Tree Society

Co-founder and President

Friends of Mohawk Trail State Forest

Co-founder, National Cadre