MHB Rectangles & Squares - Finding a Numerical Measure

[Yankel]

Hello

I am looking for a mathematical measure, that will tell me, numerically, how far is any rectangle from a being a square.
One obvious measure is the ratio between the sides of the rectangle. If the ratio is 1, it is a square. This measure is good, as it preserves a very important characteristic, which is, for similar rectangles, we will get the same measure.

I am looking for other measures such as the ratio, that will allow me to sort rectangles by how close they are to the form of a square, while preserving this characteristic of similar rectangles gets the same numerical value. In addition, is there such a measure for parallelograms ? That will tell me how far are they from a square?

Thank you in advance.

 

[Opalg]

Hello

I am looking for a mathematical measure, that will tell me, numerically, how far is any rectangle from a being a square.
One obvious measure is the ratio between the sides of the rectangle. If the ratio is 1, it is a square. This measure is good, as it preserves a very important characteristic, which is, for similar rectangles, we will get the same measure.

I am looking for other measures such as the ratio, that will allow me to sort rectangles by how close they are to the form of a square, while preserving this characteristic of similar rectangles gets the same numerical value. In addition, is there such a measure for parallelograms ? That will tell me how far are they from a square?

Thank you in advance.

For a parallelogram with sides $a$ and $b$ and area $A$, you could use the measure $\dfrac{4A}{(a+b)^2}$. That will be $1$ if the parallelogram is a square, but smaller than $1$ for any nonsquare parallelogram. Also, it will give the same measure for similar parallelograms.

 

[Yankel]

Thank you, great idea !

Can I also use angles for this purpose ?

 

[I like Serena]

Thank you, great idea !

Can I also use angles for this purpose ?

Sure.
To make a parallellogram a square, we need both square angles and equal sides.
We can combine that in one measure with for instance:
$$ (\text{any angle} - 90^\circ)^2 + (\text{longest side} - \text{shortest side})^2$$