Geometry presents us with a very important theorem called Pythagoras theorem. This theorem, named after a mathematician and cosmologist Pythagoras, gives the relation between three sides of a right-angled triangle. Learning about the pythagoras theorem we can imagine how many human beings in the past have devoted their lives in finding the hidden patterns and representing it in the language of mathematical symbols.

The relation between the three sides of the right-angled triangle has been an important breakthrough for the discovery of many other mathematical relations, and the usage of all of it in day to day life applications is exemplary. Pythagoras theorem is one such example that gives a simple relation of the sides of the right-angled triangle which has proven to be useful since then.

While Pythagoras theorem gave a relation of sides for a right-angled triangle which is a triangle with one of the interior angles equal to 90 degrees, Pythagoras theorem is useful in finding the Area of equilateral triangle since if we draw a straight line from one of the vertices to the middle-point of the opposite side, then equilateral triangle splits into two right angled triangles and thus comes within the reach of Pythagoras theorem.

Keep in mind that the sum of all the interior angles of a triangle is equal to 180 degrees; this is the universal law of triangles and valid for all triangles. Now let us imagine a right-angled triangle with the length of its sides, which are at 90 degrees to each other as ‘p’ and ‘b’ and the length of the third side as ‘h.’ As per the Pythagoras theorem, the sum of the square of two sides perpendicular to each other is equal to the square of the third side. That is, in the example, we have taken

p2 + b2 = h2


Perpendicular2 + base2 = Hypotenuse2

There are many proofs for the Pythagoras theorem, but if one remembers the rule that the triangles have interior angles as the same ratio of the length of their sides. This property of triangles, or rather the common thread among the ratio of sides and the degree of angle, has been a very useful property and gave birth to a whole new branch of trigonometry.

There is a trigonometric equivalent of the pythagoras theorem. Trigonometry deals with the relation between the sides of a triangle and the measure of its interior angle. Three primary trigonometric equations are :

Sin x = length / hypotenuse

Cos x = base / hypotenuse

Tan x = length / base

Trigonometry tells us that : square of sin x + square of cos x = 1

Let’s see if we can relate this with the Pythagoras theorem :

Square of sin x = (length / hypotenuse)2

Square of cos x = (base / hypotenuse)2

Summing them together,

= (length2 + base2)/hypotenuse2

Now, we know from pythagoras theorem that length2 + base2 = hypotenuse2

So replacing length2 + base2 by hypotenuse2 we get,

= hypotenuse2/hypotenuse2 = 1

Thus we can see how trigonometric theorems match with Pythagoras theorem.

Because of the popularity of the Pythagoras theorem and its applicability, the set of three numbers that satisfy the Pythagoras theorem are called the Pythagoras triplets. For example, numbers 3, 4, and 5 are Pythagoras triplets as the square of 3 that is 9 and the square of 4 that is 16 when summed together gives 25, which is square of 5. There are many other sets of Pythagoras triplets.

The elegance and simplicity of Pythagoras theorem are spellbinding, considering how many complex mysteries of the world get resolved using this simple key. Whether it is to find out how far are the sun and moon from earth or the creation of houses or even in the cases where triangles are not really used, it is the Pythagoras theorem, and it’s a simple relation of the three sides which does the trick.

Thus it is a very important concept to grasp and associate how Pythagoras theorem is the basis of many other formulas in geometry and trigonometry. Math worksheets related to pythagoras theorem can be downloaded for practice from Cuemath website. These worksheets will help the students gain an in-depth understanding of this important concept.

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