How to Use Trigonometry to Measure Heights from the Ground
Written on
Understanding the Basics of Height Measurement
Imagine you're about to attempt to scale a skyscraper. Before you embark on this journey, it’s crucial to know the building’s height. Unfortunately, a quick online search yields no answers, and using a tape measure is impractical. So, what’s the solution?
In this guide, I’ll explain a technique to determine the height of any tall structure from the comfort of the ground. All you need is a basic measuring tool, such as a ruler or tape measure, along with a protractor.
Let’s refresh our knowledge of trigonometry. Consider a right-angled triangle with labeled sides and angles:
Illustration by the author
Recall the relationships that exist between the angles and side lengths:
Exploring the Skyscraper Scenario
Now, let’s return to our skyscraper scenario.
Illustration by the author
In this situation, we’ve created a triangle based on various measurements. A represents the distance from the skyscraper, measured with a tape measure, and ? denotes the angle to the top of the building, which we measure using a protractor.
With known values for A and ?, we can calculate O, the height of the skyscraper, using the following formula:
This is how you can determine the height of any object from the ground. One caveat is that measuring the angle requires a bit of estimation. For greater accuracy, hold the protractor at eye level, but keep in mind to factor in your own height when calculating O.
Let's consider an example: If you stand 20 meters away from a tree and measure an angle of 60° from ground level, the height of the tree can be calculated using the formula: 20m * tan(60°), which equals approximately 34.64 meters.
Challenge Yourself
Now for a challenge: If you’ve determined that a building is 50 meters tall, how far must you walk away to measure an angle of 30°?
The first video titled "Using Trigonometry to Determine the Height of a Tall Object" explains this concept in detail and provides visual examples to help you understand the application of trigonometry in real-world scenarios.
The second video "How To Calculate The Height of a Building Using Trigonometry" further elaborates on the methods for measuring heights effectively, ensuring you grasp the techniques thoroughly.