Everyone is aware of the Empire State building or the Eiffel Tower. Historical monuments like the Great Pyramids and the Leaning Tower of Pisa are usually identified by their enormous height. But do you know how to measure the height of such a building? And the Burj: just 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 simple measuring techniques used in ancient times that still work today. The article covers three techniques, the first two use the shadow of the building, while the third uses a clinometer (protractor).

## Technique 1:

We’ll start with the simplest method, which unfortunately may also be the most impractical as it involves measuring the shadow of a tall building on the ground. 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.

All you need is a broom stick and a measuring tape. 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 that the length of shadow of broomstick = length of broomstick.

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

## Technique 2:

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 you don’t have to wait untill 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 to 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).

## Technique 3:

The method of measurement requires a protractor (clinometer), a straw and a measuring tape. This method does not use shadows. Instead it uses the accurate visual senses of the measurer. The protractor should be as large as possible for an 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 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 **additional methods of measuring the height of tall buildings**.

## References

- Making Maths: Clinometer http://nrich.maths.org/5382
- Measuring with Shadows http://micro.magnet.fsu.edu/primer/java/scienceopticsu/shadows/
- Images from mathematicsproject.blogspot.com, mathforum.org, illumination.nctm.org