BUT if we want the same perimeter (which we do) we have to take away a square for the irregular oblong to make it the same area as the regular square.
Now look the irregular oblong has less area. So we’ve proved that for rectangles. The more sides kept internal, the smaller the area. Now we desimplify the length × width equation-
ab
= ½ (a2 + b2) × ½ (a2 + b2) eventhough it would be easier to do ab, this shows what I mean.
½ (a2 + b2) makes the sides even like a square
ab
½ (a2 + b2) times it by the ratio of its real area to a squares. (like in percent)
or simply written
A = ab
How about the triangle?
This should have a bigger perimeter with the same area or a smaller area with the same perimeter as the square.
This is because they own a hypotenuse
Now in a square this is concealed within it.
We will come back to this later.
Now remember what was said earlier about squares?
Left one has less perimeter.
Because more sides are kept internal. Well this can be used on right angled triangles.
s
r
Getting the idea?
But the more r becomes different from s, the more those squares inside the triangle become disproportional. That’s why the best ratio for r and s would be 1 : 1.
Also
This has more circumference than this. this is because you always cover more of a And less area route in less Steps by going from A to B
(this can apply to everything which means I don’t
have to finish or prove anything more but I’ll do it the hard way)
This is because this
Leaves that triangle empty. And guaranteed that will be less than 2s because of
Pythagoras theoreum
(Now think about it if a square isn’t as much area when disproportioned sides- and a square is made up of triangles, what will it be when you have a triangle).
This then applies to equilateral triangles being better than all other triangles because a)
(which is why squares have more area than triangles, because of hypotenus, which if we think of right angled triangles, we shall think of it as one side being made, really, really small that its not there).
Now the hypotenuse is the biggest side of right angled triangles. This makes more added to the hypotenuse (Pythagoras theoreum), which is why equilateral triangles is best because all sides even (they are also the step up from right angled triangles).
And so when converted to a square it makes a more even square (because the hypotenuse is hidden in the square).
Remember what I said about right angled triangles non existant other side. Now we can use the idea that a more regular rectangle the more area it has.
Also what we said about right angled triangles other 2 sides from hypotenuses being more even, making more area, can also apply to equilateral triangles having more area than right angled triangles, not to mention the general pattern we can see where more regular shapes have more area. Another way of thinking about this is this (like the square), imagine.
All the squares want to become as perfect as possible ( with the fact that whatever it does influences the other 2 ‘squares’ ). Now the only solution is to become like the right, but then its no longer a triangle and also is wasting that precious perimeter (remember above about how A to B is best). Therefore our triangle will never become how it wants to be, perfectly regular. But it does do its best, an equilateral triangle which is the triangle with the most area.
Parelellograms have bigger circumference because of the exposed slopes on either side.
Trapeziums have uneven sides (and also any other quad shape not already covered) is not good either (remember about square being better than rectangle, and the squares with right angled triangle as well).
Best trapezium would be is 2 right angled triangles on each side to make an equilateral triangle on the equilateral made square, but then it would be a square.
Whats the best way to to have a square and another square? Well if we apply what was said above about the A to B thing and that this
Wastes space. The think about squares and that hidden potential.
Those sides should not be wasted. So we add a side and we get a pentagon.
* remember a refers only to that little slope not the whole side
We said the advantage of squares was their ability to hide hypotenuses of greater length. So with a pentagon you have a square and hypotenuses (look at h) concealed.
Now you see irregularity is a thief. We don’t want inconsistencies here (as we’ve already proven). So to make it more perfect a has to get more proportional to b (or the other way round however you look at it)- remember the trapezium becoming a square, so you add more sides. If a becomes more like b, that little triangle we can see on the side can hold more area (via constant circumference).
Trapezium
Now at the bottom is a trapezium. Remember what we said about trapeziums earlier, so to change it from a trapezium we add another side and so on until we get to the circle…
Now the anomaly. Circles have 1 side considered to be lots of sides or 1 side.
Now it has been shown that the best area has come from the shape being regular and having more sides. Does a circle have to be considered multi sided or 1 sided?
We know its regular as it stays constant from a point.
or in another way.
----------------------------------------------------------------------------------------------------------------------
* a = side of square
s = radius
c = perimeter
A = Area
A = π × (s × s)
= 3.141… × s2
A = a2 = 3.141… × s2
Same areas
But!
C = π × 2s
= 6.283185… × s
C = √(π × 2) × √s × 4 = 4 ( √2πs
= 10.0265131 × √s
Now that difference in 4 is nothing compared to square root which shrink huge numbers down to tiny sizes. The smaller a number you square root, the less that is taken away (bear in mind that s > 0). But we are dealing with 1000, here, not 2, so the cut would be significant.
Therefore we can say that circles should be considered multi sided instead of 1 sided.
Also we can say the more sides a shape has the bigger its area for a given circumference, also sectors of a shape being of the same shape also makes the area bigger (sectors being the center of a shape joined to all corners- including circle).
What is this area then that we get with a circle?
R = 1000 / 2π
R = 159.1549431…
A = π × (159.1549431…)2
A = 79577.47155
Lets just back check against a square.
S = 1000 / 4
S = 250
A = 2502
A = 62500
Circle is biggest. No need for number crunching to prove anything, but if you want it there are spreadsheets →Behind here is spreadsheets.
Username: alex Created on 28/03/2003 10:48 AM
