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A fun question: Logic in a geometrical concept.

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Imagine that you have a rope which is 24 cm long.

 

If you make a perfect square from this rope (each side is 6 cm) then the area becomes 6 x 6 = 36 cm^2.

If you make a perfect triangle (each side is 8 cm) from the same rope, then the area becomes (768)^0.5 which is approx. 28 cm^2

 

So the length of your border is always the same, but this same border length surrounds different amount of area for different shapes.

 

The border length is the same, but the area size is different.

 

I am not asking mathematical explanation, but I am searching for the logical explanation. What is the logic behind this? Difficult to understand the logic.

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Logical explanation I am unsure of... I would take a guess and say a circle probably gives the largest area within the rope. Something to do with larger internal angles?

 

Circumference = 24 = piD therefore D = 7.638, so r ~ 3.82

 

Area = pi x 3.82x3.82 = 45.85 cm^2

 

I'll also take a guess and say that the more sides of the shape the larger area, starting with the triangle, 3, then square, 4 - up to the circle being of infinite sides. You could test this by working out the areas for the 5 and 6 sided shapes and seeing the values fit the predicted curve. (i.e. - 5 sides, area = 39.63cm^2... so seems to be fitting.)

 

So might have something to do with number of sides compared to the internal angles of the shape.... the more sides there are the larger the internal angles are and the higher the area. The area becomes maximum as the shape approaches a circle (infinite sides).

Edited by DrP
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Logical explanation I am unsure of... I would take a guess and say a circle probably gives the largest area 2within the rope. Something to do with larger internal angles?

 

 

 

You beat me to it. The greater the number of sides of a polygon, the greater the area (for a fixed perimeter). The circle is the case of an infinite number of sides.

Edited by DrKrettin
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Logical explanation I am unsure of... I would take a guess and say a circle probably gives the largest area within the rope. Something to do with larger internal angles?

 

Thanks for the clearer mathematical explanation by using correlation between amount of sides and amount of area. However, I am still chasing the logic, we can say the physical reason instead of the mathematical reason :)

 

Are we losing area with smaller angles? Are small angles missing a smaller part of the area?

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I've done a rough calculation. See what you think. Let's draw an n-sided regular polygon where the perimeter is a constant P.

 

We can divide the area into n symmetrical isoceles triangles where the base is P/n. The half angle at the centre is pi/n so the height h of the triangle is given by tan(pi/n) = P/2n.h.

 

So the area of each triangle is P/2n. P/(2n.tan pi/n)

 

So the total area is n times that, equal to P^2/(4n.tan(pi/n))

 

I'll leave it as an exercise for the student to show that that function increases with n to a limit of (pi.P^2)/4 :unsure:

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Perimeter is the sum total of the lengths of the sides of a geometrical shape.

Area is the amount of plane occupied by the boundary of a geometrical shape.

So, it gets clear that area has no relations with perimeter.

It also becomes clear from your situation.

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I'm not sure what you mean by "area has no relations with perimeter." Certainly for a polygon with n sides there is a simple relationship between area and perimeter. And for a circle there is a simple relationship between area and perimeter. Did you mean that there the relationship between area and perimeter of a polygon changes as the number of sides changes?

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Mathematically, we have-

Perimeter of a circle = Circumference©

[math]C=2\pi r[/math]

Area of a circle(A)[math]=\pi r^2[/math]

So,

[math] A=\frac{Cr}{2} [/math]

 

But with same perimeter C, we can get an area of a square as-

side of square(s)=[math]C/4[/math]

Area[math]=\frac{C^2}{16}[/math]

 

So same perimeter C, but different areas.

 

A logical way to understand this fact is to think that area is the integration of a function under definite points. Perimeter is the functions length upto a point.

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A logical way to understand this fact is to think that area is the integration of a function under definite points. Perimeter is the functions length upto a point.

 

 

Surely the way to think is that the circumference is the differential of the area - just as the surface area is the differential of the volume for regular 3d shapes

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Imagine that you have a rope which is 24 cm long.

 

If you make a perfect square from this rope (each side is 6 cm) then the area becomes 6 x 6 = 36 cm^2.

If you make a perfect triangle (each side is 8 cm) from the same rope, then the area becomes (768)^0.5 which is approx. 28 cm^2

 

So the length of your border is always the same, but this same border length surrounds different amount of area for different shapes.

 

The border length is the same, but the area size is different.

 

I am not asking mathematical explanation, but I am searching for the logical explanation. What is the logic behind this? Difficult to understand the logic.

 

Nature always moves to the lowest energy state. If a volume of water is placed in a zero gravity environment it forms a perfect sphere which is the lowest energy state it can attain. You are observing a two dimensional abstract example of this principle, and can observe its graduated numerical representations by manipulating the boundary's shape. As the volume of the shape increases the surface ratio of angled corners will decrease in proportion. Think of a star shape utilizing as many corners and sides as possible.

 

Pressure containment and structural load bearing designs naturally take advantage of this principal so we all rather sense this in an almost subconscious expectation of the world around us.

 

 

Edited for clarity.

Edited by arc
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