The Pythagoras Theorem

The Pythagoras theorem formula describes, "The square of the hypotenuse is equal to the sum of the squares of the other two sides", which algebraically means c² = a² + b². The theorem is a special case, which works only for right-angled triangles, where a triangle has a 90° angle, and therefore it does not work on non right-angled triangles. The longest side, or the length of the diagonal, usually refers to the hypotenuse and denoted by the letter c in the formula. The other two sides, which are usually referred to as the shorter sides, are usually the letters a, or b.

This article might be useful for kids as I decided to publish this for my nephews who are studying for their GCSE exams. This general-purpose article might help just about anyone interested in the theorem. It might be useful for level 7, grade 8, grade 9, KS3, KS4, year 8, year 9, and year 10 students.

Vocabulary

The traditional statement begins with the focus on the hypotenuse, which is the result, and therefore many educators use a different way to express this theorem.

Perhaps a better definition might be, "The sum of the squares of the two sides is equal to the square of the hypotenuse."

a² + b² = c²

The statement begins with the focus on the problem and its solution resulting in the final answer. This seems more straightforward besides being even simpler. You also have a very natural ABC sequence, which requires no effort to memorise.

Children are already aware of the concept that you first "add things up" and then it equates to "something". It falls more naturally to the way we think and write, and therefore is easier to remember and understand.

In my case I can read left-to-right and right-to-left and even up-and-down in Japanese :-) so it does not make much difference, but if you were someone who never understood it at school then this might probably be the reason why. It was not your fault. I am not looking for an honorary but it would be nice if the traditional definition were more congruent with the formula in an easily understandable way for the Western mind.

Rearranged

Whilst you can calculate the length of the hypotenuse using the standard formula, you will need to learn to rearrange it to find the other sides. Whenever you need to find the length of a side that is not the hypotenuse you will have to transpose the formula to solve for the unknown side. In exams the unknown side is sometimes referred to as x and you are expected find x.

How to find a:

Use this formula to find a.

How to find b:

Rearrange it like this to find b, and then you have to take the square root of the RHS to find the final value.

Proof

The Pythagoras Theorem Proof is on a separate page as it takes up a lot of space here. For children and kids the best proof might be a visual proof or animated proof.

Calculator Solver

This is an online calculator solver to calculate the length of the hypotenuse, given the length of side a, and side b. I also have an angle calculator for finding the angles within a right-angled triangle. This is useful when you have sides and you need angles. Please use the Pythagoras Theorem Calculator if you need to know how to calculate a, or how to calculate b, or angles.

Example Practice Question

Here are some example problems found in everyday life that might be useful. These are obviously very simple questions and answers but worth going over.

Question 1

You are a Hindu Priest living in Ancient India at around 800 BC, and you have to perform the "yajna" ceremony. Unfortunately, you forget to bring your only copy of the Sulvasutra. Luckily, you go online on Google and find Peter's site, which contains the theorem.

The sides of the fire alter are required to be a perfect geometrical shape and the lengths must be multiples no less than 1. For this particular ceremony, God will be listening in on the triple frequency of 12, 35, and 37, where each figure represents a vector x, y, z, pointing into an inner sub-space dimension.

You are not sure if these figures are correct, as you have not brought your Sutra. Your job is to verify if these numbers are correct. How will you do that using the modern Pythagorean Theorem Calculator?

Answer

Enter the values a = 12, b = 35 in the calculator section above. If the answer is 37, then the frequency is correct.

Tips for educators

Here are some basic tips, which might help educators teach the theorem in a classroom setting.

How not to draw diagrams: Do not label the vertices of the triangle A, B, C, and then identify each side as AB, BC, and CA. That is confusing especially when you start teaching algebra. Instead just label each side as a, b, and c in lower case.

This theorem is about calculating the lengths of the sides of a triangle. It is not about areas! Always begin the lesson with an example of how to calculate the length of the hypotenuse. Keep the proofs and verification to the end of the class. If you start the lesson by teaching how to sum the equivalent areas for each side of a triangle then students are going to be very confused.

The article How Not to Teach Pythagoras Theorem Teachers Tips might prove useful if you need more information.

Uses in Everyday Life

You may be wondering what jobs use the Pythagoras theorem? The theorem has wide-ranging practical applications in real life. It is used today in everyday life, and without it ships would not be able to navigate, planes would not be able to fly, and humankind would not have been able to go into space. There are countless uses in real life and here is a list of some from the top of my head.

If you want to become a pilot like me, then you will need to learn this theorem by heart. Air navigation relies heavily on vectors. Vectors have magnitude and direction both of which relies on this theorem. Anyone involved in civil and military aviation, including the staff in the tower would have to know this.

If you are going to become a WhiteKnightTwo astronaut then this theorem is vital for re-entry orientation. Although the navigation computers take care of most of the maths, you will need to know this in case the computers crashed.

If you are going to be the captain of a ship then this is vital learning, otherwise you will get lost in the high seas. Of course, these days we have GPS, but in case that failed, you will have to make the course corrections manually by charting the stars, using a sextant, and this theorem.

Surveyors and architects are always using this theorem for all sorts of calculations ranging from floor plan distances to lengths of beams. They are always walking around with Theodolites, and compasses, measuring angles and distances, and the theorem is heavily used.

Astronomers use this to calculate the distances between planets in alignment and astronomical approximations of distances.

The central theme to numerology involves tabulating the "Pythagorean Square" in a 3 × 3 matrix. Numerologists use this heavily for their calculations.

History

Pythagoras of Samos was a Greek Mathematician born at around the period of 570 BC on the Island of Samos. He travelled the continent in search of "truth and knowledge".

As well as this mathematical theorem, he is the founder of the "pythagoreanism" religion. He believed that the reality as we know it could be broken down into numbers and mathematics. The central theme to his religion was the belief in the transmigration of the soul, especially its reincarnation after death.

He travelled to many distant lands. Historians have also suggested that the accomplishments credited to him may actually have been the accomplishments of his colleagues and successors; therefore, the origin of the theorem is a little uncertain.

Ancient India

Man first used this type of mathematics in Ancient India. The earliest written record dates as far back as 800 BC (and probably older) to a manual known as the Sulvasutra, belonging to one of the oldest religions of the world, Hinduism.

Sutra is another name for "manual" and there are many different types of sutras on all sorts of topics. There is only one known in the West, which became famous... However, this one is the Sulva Sutra, which means "measurement".

This manual together with many other manuals mentions the concept of measurement where the square of the hypotenuse is equal to the sum of the squares of the other two sides. The construction of temples and altars relied heavily on this type of mathematics, and the following triple figures appear repeatedly in many sutras.

• (3, 4, 5)
• (5, 12, 13)
• (8, 15, 17)
• (7, 24, 25)
• (12, 35, 37)

Written in the ancient language of Sanskrit, these manuals contained verbal mathematical instructions to make calculations for construction. Over the years, much of it has been lost, stolen, and "rediscovered".

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