Updated: Sep 4, 2021
The Pythagorean Theorem is a fundamental part of the knowledge that every student of geometry needs. This theorem has a long history as it goes back far beyond Euclid, who founded geometry as we know it today, to the mysterious philosopher-mystic Pythagoras. Despite Pythagoras’ extensive contributions to early science, maths, and philosophy, his theorem still remains his most lasting achievement. While the theorem itself only deals with right angled triangles, it gives us the ability to understand a wide range of shapes. This post will go through the fundamentals of the theorem, as well as some of the ways you can apply it in your work.
The Pythagorean Theorem
The Pythagorean Theorem allows you to know the length of one side of a right-angled triangle by knowing the length of the other two sides. As such, we need to know what to call the sides of the right-angled triangle before we can learn about the theory. The base is the horizontal side, at the bottom of the triangle. The height is the vertical side, which is adjacent to the right angle. The hypotenuse is the longest side, which is opposite the right angle.
The Theorem says that, for a right-angled triangle, the length of the hypotenuse squared is equal to the lengths of the base and the height squared and added together. In other words, if c is the length of the hypotenuse, a and b are the lengths of the height and the base respectively, then a² + b² = c².
For an easy example, imagine a right-angled triangle where the base is 3 cm long and the height is 4 cm long. From this we can work out the length of the hypotenuse. 3² cm² = 9 cm², and 4² cm² = 16 cm². To get the square of the hypotenuse we add these together: 9 cm² + 16 cm² = 25 cm² . To get the arrive at of our hypotenuse, we must take the square root of this product: √(25 cm²) = 5 cm. We have now worked out that the length of the hypotenuse is 5 cm.
It is important to note that this example is a bit unusual as the lengths of the base, the height, and the hypotenuse are all whole numbers. As such, it is an example of a something known as a ‘Pythagorean triple’. Pythagorean triples are relatively uncommon, so, in most cases that you will find in your own work, the length at least one of sides of the right-angled triangle will not be a whole number and will involve some decimal places.
It is also important to realise that any square or rectangle can be thought of as being made up of two right-angled triangles, joined at their hypotenuses. This means that the Pythagorean theorem can be used to study squares and rectangles as well. For example, if you know the length of the sides of a square, you can use the Pythagorean theorem to work out the length of a diagonal line which stretches between the corners of that square.
Ladder Practical Example
Now let’s look at some practical examples.
Imagine you are putting up a ladder in order to climb a wall. You know that the ladder is 3 m long and the wall is 2.4 m high. You want to know how far from the wall to place the bottom of the ladder so that you can reach the top of the wall. You can use the Pythagorean Theorem to help you here.
If you assume that the angle between the wall and the ground is a right angle, you can treat the height of the wall as the height (a) and the length of the ladder as the hypotenuse (c). We now need to solve for the base (b), which is the distance that the bottom of the ladder needs to be from the wall in order to reach the top.
We can rearrange the equation in the following way to solve for b.
√(c² − a²) = b
If we put in the values we already know, we are left with:
√(3² m² − 2.4² m²) = b
We can then work through the following stages to give us the answer.
√(9 m² – 5.76 m²) = b
√(3.24) m = b
1.8 m = b
We now know that the bottom of the ladder needs to be 1.8 m meters from the wall, all thanks to the Pythagorean Theorem.
Architectural Practical Application
Architects often want to know whether or not an angle between two walls is actually a right angle. They can use the Pythagorean theorem to help them with this.
For example, imagine you have two walls which are both 8 m long. An architect wants to know whether the angle between them is a right angle. She can treat these two walls as the base and the height of a right-angled triangle. She can then measure the distance between the ends of the two walls, to find the hypotenuse. If these lengths do not satisfy the Pythagorean Theorem, then she knows that the angle is not a right angle.
The architect measures the distance of the two walls and finds it to be exactly 10m.
She rearranges the Pythagorean equation to solve for the hypotenuse. She will then have the following:
√(a² + b²) = c
Yet, if the base and the height are both 8m, we can see that the hypotenuse must be 11.3m.
√(8² + 8²) = 11.3
Therefore, the measurements she has taken do not satisfy the Pythagorean theorem, and she knows that the angle at which the walls meet is not a right angle.
A good knowledge of the Pythagorean theorem gives students a firm basis from which they can go on to learn more about trigonometry. Hopefully this post has given you a good grasp of what the theorem is, and how you can use it. One key point to remember is that the Pythagorean theorem is about the relationship between the sides of a right-angled triangle. This means that any time you’re met with a right-angled triangle in your own work, the first thing that you should think is, ‘do I need to use Pythagoras here?’.
Get hold of our Exam Revision Guide and let’s turn your exam experience into a success story 😀 → https://www.studysquare.co.uk/pdf
Logic Enthusiast is an independent writer and is studying for an MA in Philosophy at the University of Edinburgh. He is particularly interested in Logic and the Philosophy of Science.