Rectangles & Squares - Finding a Numerical Measure

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In summary, the conversation discusses different measures that can be used to determine how close a rectangle or parallelogram is to being a square. The most obvious measure is the ratio between the sides, but other measures such as the area and angles can also be used. These measures preserve the characteristic that similar rectangles and parallelograms will have the same numerical value. A possible measure for parallelograms is $\dfrac{4A}{(a+b)^2}$, and angles can also be incorporated into a measure such as $(\text{any angle} - 90^\circ)^2 + (\text{longest side} - \text{shortest side})^2$.
  • #1
Yankel
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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.
 
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  • #2
Yankel said:
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.
 
  • #3
Thank you, great idea !

Can I also use angles for this purpose ?
 
  • #4
Yankel said:
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:
$$ (\text{any angle} - 90^\circ)^2 + (\text{longest side} - \text{shortest side})^2$$
 

1. What is the difference between a rectangle and a square?

A rectangle is a four-sided polygon with two pairs of parallel sides. A square is a special type of rectangle with four equal sides and four right angles. This means that all squares are rectangles, but not all rectangles are squares.

2. How do you find the perimeter of a rectangle or square?

The perimeter of a rectangle or square is the distance around the outside of the shape. To find the perimeter, you add up the length of all the sides. For a rectangle, this would be 2 times the length plus 2 times the width. For a square, it would be 4 times the length of one side.

3. What is the formula for finding the area of a rectangle or square?

The area of a rectangle or square is the measure of the space inside the shape. To find the area, you multiply the length by the width. The formula for a rectangle is length x width, and for a square it is length x length or length squared.

4. Can you find the numerical measure of a rectangle or square if you only know the perimeter?

Yes, you can find the numerical measure of a rectangle or square if you know the perimeter. You first need to divide the perimeter by 2, and then subtract the width from this result. The remaining number will be the length. For a square, you can simply divide the perimeter by 4 to find the length of one side.

5. What is the Pythagorean theorem and how is it related to rectangles and squares?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can be used to find the length of the diagonal (or hypotenuse) of a rectangle or square, if we know the length and width. This is because the diagonal is the hypotenuse of a right triangle formed by the length and width of the rectangle or square.

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