Can you believe that you can measure the height of a tall tower or a building with the help of a friend and his/her height? Read this to learn how to measure the height of a tall tower using a clinometer, a friend, and simple physics formulas.

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In my last article, we discussed three measuring techniques and the construction of a clinometer. I am sure that by now you can design your own clinometer and measure the heights of buildings using the first three techniques. However, there are three additional methods you can use which are explored in this article.

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### Technique 4:

In this method, the same clinometer is used and the procedure is also same. But an additional trigonometric idea is used to reduce the amount of work. We know that tan 45 = 1. So in this method, we fix up a straw at an angle of 45 degrees and look through the straw to locate the building top. Thus from the right-angled triangle shown in the picture attached below, it is evident that the distance the measurer is away from the building as he locates the building top, is equal to the height of the building.

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### Technique 5:

One unusual method, which may not be as accurate, is to drop a ball from the top and monitor the time taken to reach the ground. This works well if the building is tall and straight. If the building is tapered or larger at the base the method will still deliver an approximate height of the building.

You are probably familiar with this simple physics formula:

H = 0.5 * g * t^2,

Where “g" is the constant of gravitational acceleration, and g = 9.8 meters/second

^{2}And “t" is the duration of fall in seconds.

Thus we get “H" which is the distance traveled by the free-falling object, inferring the height of the building in meters.

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### Technique 6:

This method is not advisable as it would end up being a public nuisance as well as a fire hazard. However, it would work in theory.

This method could be performed for huge sky scrappers where their height substantiates the differentiation of speed of sound and light. Ask a friend to light up a cracker on the ground while you witness it from top of the building. With the help of stop watch, just calculate the time difference between the flash and boom of the cracker. Then use the formula,

H = v * t,

Where H is the height of the building in meters,

“v" is the speed of sound which is 330meters/second, and

“t" is the time duration between the flash and boom.

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### Technique 7:

This method may be called the “Yardstick" method. It requires an additional person and a yardstick. It is essential to know the exact height of your friend, as he/ she becomes the main reference, based on which the height of the building or a tall tower is calculated.

- Ask your friend to stand close to the building base or the base of the tower.
- Hold the yardstick vertically up and move back to match the height of your friend exactly to an inch in the yardstick.
- Without moving from that point and holding the yardstick in the same position, measure between the top and bottom, the number of inches the tower or the building occupies.
- Every inch equals the height of your friend. For better understanding, if your friend is actually 6 feet tall, and the tower or the building makes 50 inches in the yardstick, then the height of the tower or the building is 6 * 50 = 300 feet.

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### Technique 8:

If nothing is available and if you just require a rough estimate of the building height, just count the number of floors and multiply by 3.5 meters or 4 meters, as the average floor height is between 3.5 meters to 4.5 meters

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All the above methods are simple and do not require sophisticated equipments and technical people. They all have been used by our ancestors to determine the height of buildings. These methods would work for the Great Pyarmids too, although the calculation would be slightly different. This is because the base of the pyramid occupies a considerable area. The picture below indicates the formula used and its explanation.

It can be fun to experiment with the methods to see math in action. Today, most buildings are measured using equipment like ultrasonic transponders and radar gauges.

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*Image Credits:*mathematicsprojects.blogspot.com

mathforum.org

illumination.nctm.org

### References

- Measuring Distance & Height http://www.collierschools.com/candi/msia/documents/measuring_height_activity_complete.pdf
- Gravitation, Height & Velocity http://www.tutorvista.com/content/science/science-i/gravitation/question-answers-2.php
- Measuring Building Height - http://illuminations.nctm.org/LessonDetail.aspx?id=L764