a simple estimation of height

Geometry is useful for more than just passing the sixth grade.

In October, I posted on estimation as an essential analytical tool to have today (and more importantly, tomorrow). It’s useful for scheduling, planning, purchasing, and other decision-making circumstances. Well here’s a quick and easy geometric technique for estimating the height of very large things. All you need is an intermediate height of reference (perhaps a friend) and your eyes.

For this example, I will use a friend as my intermediate point of reference and a large building as the object for which I wish to estimate the height.

Line up your friend between you and the building. Your friend should be positioned so that when your eyes (A) are as close as possible to the ground, the top of your friend’s head (C) lines up with the top of the building (E). You’re essentially creating the hypotenuse of a large triangle!

Now, let’s label and identify the other parts of our picture.

Here are our labels.

Given this picture, geometry tells us that certain relationships exist.

Therefore, three easy estimations must be made in order to get the estimated height of the building (y):

       w = the distance between you and your friend
       x = the distance between you and the building
       z =  the height of your friend 

NOTE: Be sure to use the same units in your estimations (feet or yards, perhaps) or else your calculation will not work. 

Once you have those three values, just leave the rest to geometry. You have basically created one right triangle inside another right triangle, assuming the building and your friend are both standing up straight. Therefore they have equal angles and therefore equal ratios of their legs, allowing us to make this simple calculation. The result:

Math is fun, right? 🙂

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