# Any math experts on here? I'm looking for an equation involving rectangles...

**URL:** <https://boards.straightdope.com/t/any-math-experts-on-here-im-looking-for-an-equation-involving-rectangles/624586>\
**Category:** Factual Questions\
**Created:** [June 9, 2012, 3:36am UTC](https://boards.straightdope.com/t/any-math-experts-on-here-im-looking-for-an-equation-involving-rectangles/624586 "2012-06-09T03:36:54Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![petew83](https://avatars.discourse-cdn.com/v4/letter/p/7bcc69/32.png) [@petew83](https://boards.straightdope.com/u/petew83)\
**Post date:** [June 9, 2012, 3:36am UTC](https://boards.straightdope.com/t/any-math-experts-on-here-im-looking-for-an-equation-involving-rectangles/624586/1 "2012-06-09T03:36:54Z")

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What would the formula be to find the initial sides of rectangle with area = 1, so that the rectangle with area = x has the same side ratio? (the shorter side of this rectangle = the longer side of former rectangle)

For example, if x = 2, the initial rectangle should have sides of about 1.18 and 0.845. This gives a side ratio of 1.4-to-1 for the area = 1 rectangle, and a side ratio of about 1.43-to-1 for the area = 2 rectangle.

I want to be able to find these values for x = anything, if possible, to an accurate degree. This isn’t for school homework or anything like that; it’s a tool I need to flesh out a concept I’m pondering.

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**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\
**Post date:** [June 9, 2012, 3:47am UTC](https://boards.straightdope.com/t/any-math-experts-on-here-im-looking-for-an-equation-involving-rectangles/624586/2 "2012-06-09T03:47:53Z")

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If I’ve understood what you’re asking correctly, then your answer is that the long side of the original rectangle has a length of y = [sup]4[/sup]√x, i.e., the fourth root of x. Then your original rectangle should have side lengths of y & 1/y, while your larger rectangle has sides of length y and y[sup]3[/sup]. Note that the area of the second rectangle will be ([sup]4[/sup]√x)\*([sup]4[/sup]√x)[sup]3[/sup] = x, as desired.

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**Author:** ![petew83](https://avatars.discourse-cdn.com/v4/letter/p/7bcc69/32.png) [@petew83](https://boards.straightdope.com/u/petew83)\
**Post date:** [June 9, 2012, 4:05am UTC](https://boards.straightdope.com/t/any-math-experts-on-here-im-looking-for-an-equation-involving-rectangles/624586/3 "2012-06-09T04:05:39Z")

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Wow thanks so much. I’m amazed that you did that so quickly! This forum is the greatest.

I had to massage your answer a bit to get what I wanted, but it worked. The original area = 1 rectangle should have sides of the fourth root of x and 1/(fourth root of x).

The larger area = x rectangle should haves sides of the fourth root of x and 3(1/(fourth root of x)).

My concept of curiosity was applying this process to x^0, x^1, x squared, x cubed, etc., to basically zoom out on the original rectangle in a fractal manner. Zooming in would involve x^0, x^-1, x^-2, etc., basically breaking up the original rectangle into x equal pieces and choosing one of them for 1/x, and so on down the line. I was thinking of how fundamental 1/x is, as it appears in a lot of equations.