How to Measure A Tall Building or Skyscraper Without Leaving the Ground

Everyone must have seen, in person or in images, the Empire State building or t he Eiffel Tower. Or you may have witnessed some tall building in the city where you live. You may have also heard some one exclaiming about some huge building’s height. Historical monuments like the Great Pyramids and the Leaning Tower of Pisa are usually identified by mentioning one of the important parameters- the height. But have you ever wondered how the height of a building is measured?

And the Burj- just how how did they arrive at the height of the building as almost exactly 818 meters?

By now, you probably have a clue that this article is about the simple measuring techniques that were used in ancient ages and that are even now applicable. By using these simple techniques, one can measure height on his own, without the help of engineers or technicians- or even in the absence of measuring equipment. You can also teach your kids, when they question you about the height of these buildings. Most of the techniques are based on “Shadow Reckoning.”

This method requires patience and sunlight. It can be used if you have to measure the height of a tall tower or building where the space around the building is flat and vacant. To measure, it just requires a broom stick and a measuring tape. Just place the broom stick on to the ground so it stands erect without any support. Ensure that the broomstick is absolutely straight extending upwards. Now be patient and wait till the length of the shadow of the broomstick equals its height above the ground. Measure the length of the shadow with the measuring tape. It is obvious that at this moment, the length of the shadow of the building is equal to its height.

Re-cap:

Ensure length of shadow of broomstick = length of broomstick.

Thus, height of the building = length of the shadow of the building.

It is not always possible to use the above method due to availability of space or time. This method of measurement also employs shadows but don’t have to wait till the length of the shadow of the broomstick equals its height. Erect the broomstick onto the ground vertically. Now measure the height of the broomstick and the length of the shadow it casts. Immediately measure the length of the shadow cast by the building. By similar triangles, we can arrive at the height of the building.

To understand the property of similar triangles, refer the picture below.

The building and its shadow form a right-angle triangle. Also the broomstick and its shadow form a similar right-angle triangle. Since both the triangles are right-angled, they become similar. Thus the height of the building can be calculated from the ratios stated below.

(Broomstick’s shadow length/ Broomstick’s height) =

(Building’s shadow length/ Building’s height).

The method of measurement requires a protractor (clinometer), a straw and a measuring tape. This method does not use the shadows but uses accurate visual senses of the measurer. The protractor preferably must be as big as possible for accurate angle observation and for further calculation. The pictures attached describe the procedure to make your own clinometer. Just select a protractor with a small hole (usually provided at the origin) and tape a straw at its flat end. Also knot a string onto the protractor with a small weight at its other end. Now we can use this simple device to look at the building top to measure its height.

Procedure:

Ensure the straw is clear and try to locate the building top through the straw. The angle of elevation is evident from the string which always acts downwards because of the pull of the weight due to gravity.

Note down the distance between the building and the point at which the building top is viewed through the straw. Also note down the angle from the clinometer. The height of the building is calculated by using the formula:

Height of the building = y * tan x + measurer’s height.

Where

y = distance of the measurer from the building.

x = the angle measured from the clinometer.

It is necessary to add our height to the calculated height of the building as our reference point (eye level) is above the ground. It is not necessary to add our height if, we measure the angle keeping our eye at ground level (on which the building stands).

In the next article, we'll look at some other good methods of measuring the height of tall buildings.

Image Credits:

mathematicsproject.blogspot.com

mathforum.org

illumination.nctm.org