## Introduction: Using a Clinometer to Measure Height

In this Instructable, I'll show you how to use a clinometer to measure the height of a tall object (for help constructing your own clinometer from basic classroom materials, click here).

What you will need;

Clinometer

Tape measure

Paper

Pen or pencil

Assistant

## Step 1: Pick a Spot

Pick a spot to measure your object (I measured a telephone pole). You should be far enough away from your object that you can see the top of it, and you need to be on level ground with the base of the object. I like to set something down by my feet once I've picked my spot, so that I can easily come back to it.

## Step 2: Measure Angle

Here's where we bust out our handy clinometer. Look through the straw of your clinometer at the top of the light pole (or whatever object you're measuring). The weighted string should hang down freely, crossing the protractor portion of the clinometer. Read the angle shown, and subtract from 90° to find your angle of vision from your eye to the top of the pole (it can be helpful here to have an assistant to read the measurement while you look through the straw). Record your results on your paper.

From my spot, my clinometer (read by my assistant) showed 55°. Subtracting from 90°, that indicated that I looked at an angle of 35° to the top of the telephone pole.

## Step 3: Measure Distance

Once you have your angle of vision, use your tape measure to find the distance from the spot you're standing to the base of the object you're measuring (an assistant comes in handy here, too). We must know how far away you are to accurately calculate the height.

My spot was 15.6** **meters from the base of the telephone pole I measured.

## Step 4: Find Your Eye-height

The last piece of data you need to calculate the height of your object is the height from the ground to your eye (your eye-height). Have your assistant help you measure this using your tape measure.

My eye height was recorded for this example as 1.64 meters.

## Step 5: Draw a Picture

Time to move inside. In calculating the height of the object you just measured, I find it helpful to begin by drawing a picture and labeling it with all of the information I have.

## Step 6: Model As a Triangle

The next step is to simplify your drawing to model your system as a right triangle. Label your triangle with the angle you read on your clinometer as well as the distance you were standing from the object (we don't need the eye-height just yet).

## Step 7: Solve for X

We can find x in this triangle (which represents the portion of the height from eye-level up) by using some basic trigonometry, specifically the tangent ratio of the triangle:

tan(**angle**) = x / **distance**

Multiply by the distance on both sides and you get:

x = tan(**angle**) * **distance**

Use a calculator to multiply these together and get a decimal value (be sure your calculator is in 'degrees' mode, rather than 'radians'!).

In my example:

tan(**35°**) = x / **15.6**

x = tan(**35****°**) * **15.6**

x = **10.92** meters

## Step 8: Combine With Eye Height

To find the height of your object, bring this x value back to the original drawing. By labeling it, we can see that the height of the object, h, is equal to the x value we just found plus the eye-height we measured earlier:

h = x + (**eye-height**)

In my example:

h = **10.92m **+ **1.64m**

h = **12.56m**

There you have it! A few basic classroom materials and a little bit of trigonometry and you can measure the height of anything around you!

I used my phone with a straw on top and the compass app (the apple compass has a level, which is nice)

another thing i noticed is that if you keep walking towards the object until the clinometer says 45 degrees, the distance between you and the object is exactly the height of the object (+ your eye height)

great but should the eye height taken be always

Unless you place your eyes close to the floor, yes, the height is necessary. in this example there's a 1.64 m difference. Unless you're measuring the height of Mt Everest, the error is probably too significant to ignore.

Thank you very much. With out this it is impossible to complete my math activity

THANK YOU SO MUCH

it was very helpful for my school project!

use pithogres thrim a2+b2

Fantastic! Thanks a lot!

I used this in class with my students it worked great!

Great instructable. im glad to see this tool on the site.

People have been mentioning the BSA method, which is fine, but the accuracy compared to a protractor and tape measure will be much less. And if you don't need accuracy, why even use the stick, just eye-ball it. (My two cents)

BSA handbook has an easier method that doesn't require an assistant or math or a protractor. Just cut a stick at eye level push stick into ground and lie down with feet against stick. When top of object is level with top of stick, mark where you eye is and measure to base of the tall object.

Ok great, make an instructable.

There is a little mistake in your logic , formula

4. must be H=P(sin phi)

5. mus be X=P(cos phi )

try it with a value phi = 85 and you will see when you trace it on scale

Shows how long it's been since I used any algebra on purpose, but how did you get the 35 degrees for the angle?

I'm not seeing where you used the eye height in your calculations. Did I miss something?

We added the eye-height to the height (x) we found in the Tangent calculation to get the final height.

Check out step 8.

If you are in Britain, the clinometer shown here is called a protractor available at most stationers, this method only works on level ground, more maths if your on a slope

The green plastic protractor is the main piece of equipment here, but clinometer is the name we used for the entire device when assembled (made up of protractor, straw, string, and weight).

A few years ago (taking a Trig review course at the time) I did something similar. My approach was more expensive than yours; my goal was best possible accuracy.

Coincidentally, I measured a telephone pole too.

Rather than making eye-level measurements, I made ground-level measurements, and measured the angle of elevation at night (since I was using a laser pointer for aiming).

The attached photo shows a level with integrated laser pointer, and a 360-degree protractor.

What if it is a very distant object (radio tower on a hill)? How would you get the horizontal distance and thus complete the calculation for its height?

How would one go about doing this without a calculator?

To do this without a calculator, it'd be handy to remember that a 45º right triangle has equal sides, and a 30-60-90 triangle has a 1:2 relationship between the sides opposite the 30º and the 60ºangles.

1) Focus on a point on the object (let's call it point A) and walk back from the object until the inclinometer reads 45º. The horizontal distance from the point on the ground directly under Point A to where you are standing, PLUS the height of your eye-level (because that's where the inclinometer is measuring 45º from) is the height of the point you are measuring.

2) Do the same thing, but at 30º, multiply the ground distance by 2 and at 60º divide the ground distance by 2. (Add the "eye level height" BEFORE you multiply/divide)

I miss the EDIT function... the 30-60-90 part is incorrect.

You're right about the 45º triangle, but the 1:2 relationship in the 30-60-90 triangle is between the short leg and the hypotenuse. To use that triangle, you would have to multiply the ground distance at 60º by sqrt(3), and divide by sqrt(3) at 30º, neither of which is easy without a calculator.

dmuldoonlla's sketch of the 60 30 right angled triangle should show the root3 on the hypotenuse. Root 3 is 1.732

On 45 degree triangle it works out at root 2 which is 1.414.

So, soon as you see a 45 triangle you know instantly that the hypotenuse is 1.414 (whatever units you are using) long

same with a 60 - 30 being 1.732 units long.

Ratio of sides....had it drilled into us in technical college, can't seem to forget it now.

I'm afraid you're incorrect. The sqrt(3) term is on the long leg, not the hypotenuse (I'm a geometry teacher, I go through this every year)

http://mathworld.wolfram.com/30-60-90Triangle.html

http://en.wikipedia.org/wiki/Special_right_triangles

.

If you're interested to see why the hypotenuse is 2* the short leg, begin with an equilateral triangle (all three sides the same). If you then cut this triangle in half vertically and look at just one half, you will have a 30-60-90 triangle, in which the hypotenuse (one of the original sides) is twice as big as the short leg (half of one of the original sides).

Yep, you're correct. I got that and the 3-4-5 triangle confused... 45-45-90 it is.

The trigonometric functions (like any other function) can be drawn and measured physically with a ruler. If you draw a line at ßº in a circunference of radius=1, the segment that is tangent to the circle in the x axis and reaches that line is the tangent function (green). The bigger you draw the circle the more accurate result you will get.

Very interesting now to get Pythagoras straight in my brain after half an cetury

Now teamwork is good, but how can I accurately read the clinometer without an assistant ?

Look through the clinometer until you see what you're supposed to see, wait for the string to stop moving, then pull the string tight against the protractor, turn it, and look at it.

Or, you could adjust your position until you get a 45° reading and then, the measured distance on the ground plus the height of the observation point would give you the height of the post without needing a scientific calculator or a sine table. This is an old, established method taught to boy scouts.

Agreed. Learned this when I was 9 or 10... And I HATE math.

Love your Instructable! I am going to use it today to measure height of a broken tree limb in our back yard. Nothing wrong with your math now, you must have made some changes. The one thing I remembered from my geometry class was "the side opposite a 30 degree angle is 1/2 the hypotenuse." Interesting that this was caught by the world's greatest living physicist, Stephen Hawking (post below).

Interesting, but it gave me a headache

It looks to me from your diagram as though the side opposite the 30 deg angle has a 1:2 relationship not with the side opposite the 60 deg angle but with the hypotenuse.

Well, if it isn't my old nemesis math .. we meet again!

This takes me back, I can remember doing something like that in grade 9 or 10 math class. The teacher threw in a bit of a twist, the object whose height we we measuring, a tall chimney, was inaccessible inside a fenced off area so there was no way to measure ones distance from the base of the chimney. The teacher asked us to calculate the height of the chimney and it's distance from the fence. This required us to take sighting from two spots outside the fence a measured distance apart. Then use the law of sines to calculate the length of the hypotenuse of the more distant sighting then use that to calculate the height and length of the base.

There is a simple method to measure heights. Needs an assistant and a 12 inch ruler or 30 cm ruler.

Using a twelve inch ruler held at arm's length, align the 12 inch mark with the top of the object and zero inches with the ground level at the foot of the object (adjust distance as required). Now get your assistant to mark the object with chalk or similar, or hold their finger at height of the one inch mark (commands such as up a bit, down a bit may help). Measure the height of the chalk mark or finger in inches, this is the height of the object in feet. Clearly only works to heights that one's assistant can reach, so around 80 feet or so. For taller trees, use the half inch point, this gives half the height in feet. For users of the metric system, a 30 cm rule and 3 cm would give one tenth the height at the mark.

There is also the Boy Scout - "stick and thumb" method: From where you are standing, take a stick and view the object so you have the top of the object at one end and your thumb marking the base... then turn the stick sideways so the stick now measures to an identifiable object on the ground... now walk off the distance between the base of the object to be measured and the 'target'...

I had a toy that did this in the late '70s, it even had a chart on the back for the trig.

my good compass has cotangent tables on the back for navigation purposes, same thing but on the horizonatl plane

nice idea bro :)

Nice !

Also note that fzumrk and hlanelee are elegant solutions.

Would the three of you be kind enough to post more for those of us who lost all those school applied maths somewhere in adulthood ??

It is so much fun to feel one can be (somewhat) clever again …

Thank you again !!!…

Another way to measure tall objects with only a tape measure is to measure the length of its shadow. This is dependent on the weather being sunny and the terrain surrounding the object being relatively flat and accessible, so its not going to work in all situations.

In your pictured example, you would measure the length of the pole's shadow. Then you need the height and shadow length for another object for comparison. You could use one of the nearby pipe bollards. Once you have the height and shadow length for the pipe bollard, the pole's height = (bollard height / bollard shadow length) * pole shadow length.

Hold a stick in a vertical position so that the distance from your fist to the top of the stick is about the length of your forearm. It makes about a 45 degree angle and the tangent is one (opposite = adjacent). Since like triangles have a common ratio, when you sight on an object with the base lined up with the top of your fist and the top of the object lined up with the top of the stick, then the distance that you are away from the object = the height of the object.

Real practicle lesson of Trig. Don't stop bringing applied math.

With my appreciation.