### Important uses of Pythagoras Theorem in Nature

Pythagoras Theorem is one of the useful geometrics in the history of mathematics and physics. Which is name after the famous Mathematician in ancient period **“Pythagoras”**

It was invented as known by the **Pythagoras** since 1st century 543 B.C , now it is used by many physicists and mathematicians in measurement of a vast distances.

##### The theorem stated that: **in** **every** **right** **triangle** **the** **sum** **of the** **square** **of** **the** **two** **sides** **(****base** **and** **perpendicular)** **are** **always** **equal** **to** **the** **square** **of** **the** **third** **side** **(hypotenuse)** **of** **the** **given** **triangle.**

**in**

**every**

**right**

**triangle**

**the**

**sum**

**of the**

**square**

**of**

**the**

**two**

**sides**

**(**

**base**

**and**

**perpendicular)**

**are**

**always**

**equal**

**to**

**the**

**square**

**of**

**the**

**third**

**side**

**(hypotenuse)**

**of**

**the**

**given**

**triangle.**

**c^2** = a**^2** **+ b^2**

It is used by some Mathematician which it leads them to the invention of trigonometry to measure the distance of the Planets, Stars etc.

By the used of this theorem in 425 B.C, **Aryabhata** studies the ratio of these three sides of the Theorem and arrange it in such a way that it links with the angle between the arms of the hypotenuse and the base of the right triangle. During his studies about the moving of the planets in the Orbit.

Here are some ratio of trigonometry of right triangle with angle **A** and its sides are hypotenuse **h,** base **b**, perpendicular **p **given below.

**p/h = sin (A)****b/h = cos (A)****p/b = tan (A)****h/p = cosec (A)****h/b = sec (A)****b/p = cot (A)**

Later **F.J.H Wollaston** invented **Hypsometer** which is base in the trigonometry the instrument used to measure the height of the tower and the distance from the observer.

#### What is Hypsometer?

*A* *simple scale hypsometer allows the height of* * a building or thees to be measured by sighting across* *a* *ruler to the base and top of the object being measured, when the distance from the object to the observer is being known.*

Modern hypsometers use a combination of laser rangefinder and clinometer to measure distances to the top and bottom of objects, and the angle between the lines from the observer to each to calculate height.

An example of such a scale hypsometer is illustrated here, and can be seen to consist of a sighting tube, a fixed horizontal scale, and an adjustable vertical scale with attached plumb line. The principle of operation of such a scale hypsometer is based on the idea of similar triangles in geometry.

#### How does it works?

First the adjustable vertical scale is set at a suitable height.

- Step 1 in the illustration, a sighting is taken on the top of the object whose height is to be determined, and the reading on the horizontal scale, h’, recorded.
- Calculation from this value will eventually give the height h, from the eye-line of the observer to the top of the object whose height is to be determined.
- Similarly as in step 2 of the illustration, a sighting is taken on the base of the object whose height is to be determined, and the reading on the horizontal scale, d’, recorded.
- Calculation from this value will eventually give the distance from the base of the object to the eye-line of the observer.

Finally the distance x from the observer to the object needs to be measured.

You know it is very interesting to know that during ancient periods Astronomers measure the distance between planet *Earth* and the other *objects* in the space by using **Parallax** Method.

#### What is Parallax?

*Parallax is* *a displacement or difference in the apparent position of an object viewed along two different lines of sight of your eyes to that point of objects that you looking for, and is measured by the angle or semi-angle of inclination between those two lines.*

Because the eyes of humans, and many animals, are located at different lateral positions on the head, binocular vision results in two slightly different images projected to the retinas of the eyes.

The differences are mainly in the relative horizontal position of objects in the two images. These positional differences are referred to as horizontal disparities or, more generally, binocular disparities. Disparities are processed in the visual cortex of the brain to yield depth perception.

Many more things Scientist and Astronomers used the trigonometric geometry to the field of their studies, to overcome the accurate measurement from the tiny length to the vast distances.