Is Galileo's square-cube law a universal proof of similarity in 2D shapes?

[adjacent]

Homework Statement

We know the if two 2D shapes are similar, the ratio of their areas are equal to the ration of the square of the corresponding sides.
$\frac{A_2}{A_1}=(\frac{l_2}{l_1})^2$

Prove this

The Attempt at a Solution

For rectangles:
Let the area of first rectangle be $A_1$ and the 2nd rectangle be $A_2$.
Let the sides of 1st rectangle be l and b and the second rectangle be $l_2 \text{ and b_2}$.
1st rectangle......Second rectangle
Area=lb......Area= $l_2b_2$ we know that $l_2$ is kl and $b_2$ is kb
So $klkb=k^2lb$
Which means that Area of larger rectangle is $k^2 \times$area of smaller rectangle.
Area=$k^2lb$
Therefore, $l_2b_2=k^2lb$ So $k^2=\frac{l_2b_2}{lb}$(in other words $\frac{A_2}{A_1}$
So $\frac{A_2}{A_1}=k^2$ and we know that $k=\frac{l_2}{l_1}=\frac{b_2}{b_1}$

So $\frac{A_2}{A_1}=(\frac{l_2}{l_1})^2$

But how do we know that this is general for ALL the 2D shapes?

 

[mfb]

How did you define "area" and "2D shapes"?
This is a non-trivial question as the validity of the statement (together with ways to prove it) depends on those definitions.

 

[adjacent]

A two-dimensional figure, is a set of line segments or sides or curve segments or arcs, all lying in a single plane.
They have length and width but no thickness.

-Area is the amount of space inside the boundary of a flat (2-dimensional) object.

 

[Ray Vickson]

A two-dimensional figure, is a set of line segments or sides or curve segments or arcs, all lying in a single plane.
They have length and width but no thickness.

-Area is the amount of space inside the boundary of a flat (2-dimensional) object.

This "definition" is not a useful statement; it just puts the same undefined concept in different words and so is not really a definition at all. What, exactly, is meant by "amount of space"? How do you measure it? This question is 100% serious!

 

[adjacent]

What, exactly, is meant by "amount of space"? How do you measure it? This question is 100% serious!

-Well, I think this is very difficult for me but I will try to do my best.I am a Gr.10 student,so please bear with me
The SI unit of area is $m^2$. Which means that two linear dimensions are multiplied together.
If we take a line, the space it covers is the length.(Maybe it covers no space at all because height is zero?)
Can you give me a hint?

 

[mfb]

That does not even cover cases like the attached figure. And there are much more problematic objects.

If you did not get a proper definition of "area" in class, you have to do some hand-waving, as there is simply no definition you could refer to. As an example, you can start with the formula for the area of a triangle, and show that this scales in the right way - and then try to find some way to express your "shapes" as a set of triangles.

 

[adjacent]

For circles too? I will have to draw many triangles to approximate the area of a circle.

 

[micromass]

Where did you find this question?? Is this really something you need to solve for school?

I really doubt that you can prove this if you don't know any calculus. Sure, you can show it for special figures like rectangles or circles, but not in general.

 

[adjacent]

Where did you find this question?? Is this really something you need to solve for school?

I never post my actual homeworks here.
My teacher was revising similarity yesterday and this question came to my mind. But he didn't give any satisfactory answer. He said: The area is already squared so the sides should be squared too to balance the equation..

I really doubt that you can prove this if you don't know any calculus. Sure, you can show it for special figures like rectangles or circles, but not in general.

What calculus do we need here? Integration?

 

[SteamKing]

For circles too? I will have to draw many triangles to approximate the area of a circle.

Doesn't a circle enclose a 2-D area?

 

[micromass]

What calculus do we need here? Integration?

I fear you need more than calculus. You will need a satisfactory theory of "area". This is done in measure theory. Then you also need to have a satisfactory theory of "length of a curve" which also brings us to measure theory.

The idea of a general definition of area is simple: first define area for rectangles in the usual way. Then cover any area by rectangles (possibly infinitely many) and take limits. This is the method used by the Greeks "the method of exhaustion" and can be used here also to get a good definition for area.

Getting a good definition of length is a bit more difficult, unless the perimeter of the curve is nice enough.

 

[adjacent]

I fear you need more than calculus. You will need a satisfactory theory of "area". This is done in measure theory. Then you also need to have a satisfactory theory of "length of a curve" which also brings us to measure theory.

The idea of a general definition of area is simple: first define area for rectangles in the usual way. Then cover any area by rectangles (possibly infinitely many) and take limits. This is the method used by the Greeks "the method of exhaustion" and can be used here also to get a good definition for area.

Getting a good definition of length is a bit more difficult, unless the perimeter of the curve is nice enough.

aaah. :grumpy: I wish some magical being comes and teaches me all these.

Doesn't a circle enclose a 2-D area?

: When did you become a mentor? Confratulations!
Yes. Circles are 2-D shapes.The problem is drawing infinite triangles. So I can't solve this problem without further knowledge?

 

[SteamKing]

I never post my actual homeworks here.
My teacher was revising similarity yesterday and this question came to my mind. But he didn't give any satisfactory answer. He said: The area is already squared so the sides should be squared too to balance the equation..

What calculus do we need here? Integration?

2-D areas need not be enclosed by 3 or more straight lines. As was pointed out, circles enclose 2-D areas, so do their cousins, the ellipses. There are many examples of 2-D areas which do not contain any straight lines whatsoever, being enclosed entirely by curves or one form or another.

 

[mfb]

For circles too? I will have to draw many triangles to approximate the area of a circle.

You can still take the limit for an infinite number of triangles, as micromass explained.

As an even more hand-waving argument, the area of every "reasonable" shape is given by the multiplication of two lengths, with an additional constant factor to account for the shape. This constant factor is constant for similar shapes, and the product of two lengths goes with the square of the scaling factor.

 

[BruceW]

yeah, as others have said, to 'prove' this in the rigorous mathematical sense would require some strict definitions, I think.

Having just looked on wikipedia, http://en.wikipedia.org/wiki/Similarity_(geometry ) http://en.wikipedia.org/wiki/Square-cube_law They seem to call this Galileo's square-cube law.