Perhaps, you have memorized the 2,500-year old Pythagorean theorem as a series of letters and numbers: a²+ b² = c². If so, then you also know that it concerns triangles that are right-angled. But I want to inform you today that, it does not concern triangles alone, it concerns all plane shapes. Hold on, I will explain this shortly.



Quick Review on Areas


The following is an excerpt from previous lesson about similar shapes and their areas.

When we were taught areas in school, there is an insight that is usually overlooked. The fact is that the area of any shape can be calculated from any line segment squared. In fact, it can be represented as:

Area = area factor × [line segment]²

Area = f × [line segment]²

Consider a square, if we are to take the length of a side s as the line segment, then the area of the square is s².




Area = area factor × [line segment]²

Area = s² = 1 × s²

If we are to use the diagonal d of a square as the line segment, the formula for the area changes to:




Comparing the two formulars above butresses the point that area of a shape can be calculate from a "line segment" squared. The only difference in the two formulars is the "multiplier", that is called the "area factor",f. Using the length of the side of a square the area factor is 1. While using the diagonal of the square, the area factor is ½. This means the area of the shape can be represented as
:


Area = area factor × [line segment]²

Area = f × [line segment]²


The following table shows different shapes, their are formular and area factor.



Intuitive look at at the Pythagorean theorem

Any right triangle can be divided into two other similar right triangles:



Now, it's obvious that the area of the two smaller triangles add up to the big one:


Now, from a previous lesson about similar shapes and their areas, we learnt that we can represent the area of a right angle triangle as:

Area = area factor × [hypotenuse]²

Area = f × [hypotenuse]²

From that post, we also learnt that similar shapes have the same area factor.

Now, from the diagram of the similar triangles above, it follows that.

Area(Big) = Area(Medium) + Area(Small)

F • c² = F • b² + F • a²

Divide all through by F:

c² = b² + a²

Yuppy! That's our famous Pythagorean theorem. Now you know intuitively that it's true because:

• A right triangle can be split into two similar right triangles</li>
• The square of the hypotenuse of a right triangle determine its area</li>
• Since the areas must add up, the squared hypotenuses (which determine
area) must add up as well.

One can visualize the Pythogerean theorem by drawing squares on the sides of the a right triangle:



The catch: You can use any shape!

Do you still remember I stated that the Pythagorean theorem can be used for any plane shape? Now, let me prove my point!

Assume that the sides of a right angle triangle represents the radius of a circle:



Now let me add the areas of the circles together:




π • 5² = π • 4² + π • 3²


What do we get when we divide through by π? Yea, you are right;

Circle of radius 5 = circle of radius 4 + circle of radius 3.

That means that the amount of paint that will be used to paint a circular wall of radius 5 will paint two circular walls of radius 4 and radius 3! Do you now agree that the Pythagorean theorem is awesome?

The fact is that, we can multipy the numbers in our Pythagorean theorem with any area factor and the theorem will still hold!

More Examples!
Surface Areas:

The surface area of a sphere is 4πr². For similar spheres the area factor is constant. It follows that in terms of surface area:

4π • 50² = 4π • 40² + 4π • 30²

<b>Area of radius 50 = Area of radius 40 + Area of radius 30</b>

Now suppose plastic balls are to be covered with leather, this means that the amount of leather that will be used to cover a ball of radius 50cm will cover two balls of radius 40cm and radius 30cm!

Physics
In Physics the energy of a moving body is given as ½mv², it follows that:

<b>Energy at 500m/s = Energy at 400m/s + Energy at 300m/s</b>

It means that, With the energy used to accelerate one bullet to 500 m/s, we could accelerate two others to 400 and 300 m/s.

Wonderful Insight

We were taught in school that the Pythagorean theorem applied to triangles, but now we know that's not true. We now know that it can apply to any similar plane shapes. In fact, it can apply any relationship between quantities that involves a parameter squared as in the example of the energy of a moving body above.

Now, go forth and enjoy your new insight!

Happy Math.

why are pictures not showing na..

I'm not a maths guru but was the theorem really from black people?

nairalandposter:
I'm not a maths guru but was the theorem really from black people?

Yes.