## 24 Aug Why Learn Pythagoras’ Theorem?

Here is a fun fact that you probably did not know. Like any other famous concept or celebrity, the Pythagoras’ Theorem was shown on the Emmy Award winning Simpsons. Given that math is not a subject you’d expect to see on your comedy program, you may have well missed this.

## Pythagoras’ Theorem in the Simpsons

In the 5^{th} season of the Simpsons, this famous geometry concept was alluded to. In the scene, Homer Simpson tries on a pair of spectacles and proceeds to recite a formula:

“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

Following which, he is corrected by a voice:

“That’s a right triangle, you idiot!”

While the Pythagoras’ Theorem is not explicitly named, by subbing in the correction, you would have its formula.

*Photo Credits: The Simpsons*

We cite the example of the Simpsons to demonstrate how well known the theorem is. This is because of its importance as a foundation in mathematics and its wide range of applications. To help you discover its usefulness, in this article, we will be looking at:

- What is the Pythagoras’ Theorem?
- Simple applications of the Pythagoras’ Theorem
- Real life applications of the Pythagoras’ Theorem

## What is the Pythagoras’ Theorem

The Pythagoras’ Theorem is a geometry statement that proves the relationship between the lengths of a right triangle. Its formula is as follows:

a^{2} + b^{2} = c^{2}

Based on this statement, you can find the length of a third side of the right triangle when provided with the first two sides. As such, you could use the theorem for solving a host of different construction and navigation problems.

This part of the concept should be familiar for most of you who have learnt the theorem in school. Yet, Pythagoras’ Theorem is in fact applicable for concepts beyond that of the right triangle.

### Application to Other Shapes

We start this section with a simple assertion. The Pythagoras’ Theorem can be applied to any shape or formula which squares a number. Why is this so? The answer lies in how many shapes can be split into multiple right triangles.

An easy application of this idea is that of a square or rectangle. By splitting a square diagonally, 2 equal right triangles are formed. As such, you can now proceed to determine the length of the diagonal, a previously unknown variable.

More importantly, a triangle can always be split into multiple right triangles. This triangle splitting allows you to split quantity (c^{2}) into 2 smaller quantities (a^{2} + b^{2}), which are the sides of the right triangle.

## Advance Applications – Conservation of Squares

Remember how we said that Pythagoras’ Theorem could be applied to any formula that squares a number?

By taking the length of a side to be a variable such as distance, energy, time or even number of people, you can calculate these missing values. In other words, while not related to shapes, if a concept has a formula that squares a number, its variables can be calculated. Here are some examples.

### Value of a Network

The value of a network attempts to calculate the worth of your network. It is based on Metcalfe’s law which states that the value of a network is n^{2}. Given that Metcalfe’s law involves a squared number, you can use Pythagoras’ theorem to determine the value of multiple networks.

For example, if you have:

- A first network with 40 people
- A second network with 30 people

Then the value of your combined networks would not be equal to 70 people. Instead based on their square values, the combined value of your network would be equal to a single network of 50 people.

Value of 50 People Network = Value of 40 People Network + Value of 30 People Network

### Energy for Accelerating an Object

Consider the amount of energy or fuel required for cars to run at a particular speed. Like before, you might think that the amount of energy required to accelerate a car at 50km/h would equal to that of 30km/h and 20km/h combined. Yet based on Pythagoras’ theorem, we know this not to be true. Instead, you would need the amount of fuel required to power cars at 40km/h and 30km/h.

Energy for car driven at 50km/h = energy for 40km/h + 30km/h

Based on this understanding, we can see that the amount of fuel consumed for faster speeds does not scale linearly. Instead, as the speed of your car increases, it requires more fuel per speed unit added.

From another point of view, if you were in charge of a train transportation company, you would realize the following. For the amount of energy needed to power a train to reach 50km/h, you could power 2 other trains at 40km/h and 30km/h respectively.

## Common Real Life Industry Uses

### Navigation on land, air and sea

Pythagoras’ Theorem is the basis for many navigational and topography techniques. The premise of which is fairly simple.

For example, you might wish to navigate to a point that is 30 miles south and 40 miles east. By taking these 2 measurements as the lengths, you can then find the hypotenuse of the triangle which is the shortest path to the destination. In doing so, you would find its exact bearing and the approximate length needed to be travelled.

Another variation of the application of the theorem lies in the descent of an airplane. Instead of a horizontal triangle in the previous example, imagine a vertical one this time. As a pilot approaching an airport, you can use the height of the plane from the ground as 1 length. The other length would then naturally be the distance to the airport. Using these 2 lengths, the hypothenuse can be calculated and thus inform you when to begin your descent.

### Architecture

Architects are often challenged to imagine the exact dimensions of a space or structure that is currently non-existent. As such, they use Pythagoras’ Theorem extensively to correctly determine the length of materials required.

For example, when looking to install a slope roof, architects need to determine beforehand the dimensions of the roof. This is made possible by finding the length and breadth of the space followed by calculating the diagonal. With all measurements on hand, the correctly sized beams and roof can be installed.

### Surveying

Cartographers carry out surveying in order to create a map. Yet in order to do so, calculations of distances and heights need to be made. These help to populate the map and inform the user of elevations, spatial distances and the general geography of the area.

Yet, for many natural environments, it can be difficult to determine the length and gradient of their slopes at each portion. This is where Pythagoras’ Theorem comes in handy. A surveyor would use:

- A telescope from a fixed distance away
- A measuring stick of known height

The distance and height then form the two lengths of a triangle. The slope of the terrain then naturally becomes the hypothenuse and thus can be determined using the theorem. Information that can be inferred include the slope’s length and steepness.

## Einstein’s Takeaway:

As seen in this article, the Pythagoras’ Theorem while not being immediately tangible, is in fact highly useful in real life. Furthermore, it is widely used in order to calculate intermediate data which in turn is used to infer more data.

Given its intangibility, many students may struggle to utilize it effectively in situation based questions. This is where **sec 3 E math tuition** and **sec 3 A math tuition** are valuable. Having the extra attention by qualified tutors will allow you or your child to fully grasp the Pythagoras’ Theorem.

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