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? 🙂

life optimization through estimation

The ability to accurately estimate a target value is an asset to any brain. Learn to hone this ability, embrace it, and use it to optimize your life.

Our lives are surrounded by invisible data – most of it in units of time, energy, space, and money. Essentially, our brains are huge folded databases that store this data, and use it to make decisions, plan ahead, and live each day. But as with many types of data, there exists some uncertainty about that data. Unknowns about how long, how big, how much, from where, until when, should i, almost enough, maybe tomorrow… well you get the picture. Our life data is filled with unknowns.

That’s why estimation is essential. Without it we’d get lost, fall behind, and lose our sense of security and awareness. Whether we know it or not, our brains constantly work to estimate and approximate values, given set of life data at that moment in time. And whether we know it or not, our brains run predictive models to assess hypothetical scenarios, basically using present life data to predict future life outcomes.

These are important realizations, and strong connections of human nature to an innate mathematical realm. Estimation is both an art and a science, as it takes creativity and thought supported by various numerical methods. Having the mathematical ability to estimate proves useful in most situations, but without the artistic component, you lose the ability to understand and contextualize your estimation.

The main point here is that estimation should be embraced as part of human nature, supported by numerical methods. This is how we can optimize our life – by recognizing the units with which our lives are measured each day, and reducing as much uncertainty in those values as humanly possible. It will not make you completely successful and happy and secure, but it will get you close.


Here are some random examples of estimation from my life. The methods of estimation vary, but the fundamental questions being asked all have outcomes of an unknown nature.

1. Shopping: Budgeting $150 for a dinner party, break budget down to categories of purchases then allocate funds accordingly. Estimate totals and percent of total budget category to make decisions on necessity.
Outcome: Go bigger on the dinner and ask a couple guests to bring desserts.

2. Sports: Ten minutes left in the game, down by 2 goals. Have two full lines of players so will sub soon and again with 4 min left. Need at least 1 goal every 4 minutes leaving a 2 min buffer to protect the tie and go for a win, should allocate 60% of strategy to offense and 40% to defense for next 8 minutes. If I’m in for 6 min and need 60% offensive mindset, how inclined should I be to make a run towards the goal, leaving my defensive position?
Outcome: Win

3. Personal Finance: How much to take out at the ATM? Need to estimate expenses for the week – lunch, happy hour, gas, dinner, cab to meeting, etc. How often will I use my credit card? Am I more inclined to spend if I have cash? Will I be near another ATM this week if I need more cash? How conservative should I be in my spending given the holiday season is arriving?
Outcome: Take out $60 and bring lunch.

4. Daily Planning: Got a hour-long meeting at 3:30pm, soccer game at 6:30pm. Assuming there will be traffic, it will take me 35 minutes to get home then 5 minutes to change, 10 to heat up leftovers, 10 to eat, and 15 to switch and fold laundry. Need 25 minutes to get to field and 15 min to warm up. Will I have enough time if my 3:30pm meeting goes long or do I need to put off the laundry and/or dinner?
Outcome: Always put off laundry, but never dinner 😉


Estimating how much gold there is in the entire world
Estimating how much money there is in the entire world
Estimating the height of anything using geometry
A bit about estimation in statistics