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When pricing paintings artists often multiply a factor (F) by linear size (height + width) opposed to area (height x width) thus avoiding an extreme disparity in price between small and large paintings. (The multiplying factor will determine the monetary amount for all the paintings)

A painting of 10" + 16" multiplied by a factor of '40' = $1040

Simplified example below:

20"+40" = 60 and 40"+60" = 100 - giving a reasonable 40% price difference between prices.

20"x40" = 800 and 40"x60" = 2400 - a staggering 300% difference in price.

However using this linear size pricing method gives rise to another problem:

Lets say we had two paintings, one 20"x40" and the other 30"x30".

20+40 = 30+30 so in this case both paintings would be calculated at the same price (60 x F), despite the fact that one painting is actually 10 square inches larger in area.

So when using the linear size instead of area for price calculations clearly the result would also need to be multiplied by a greater amount the squarer a painting is.

I'm wondering if some sort of calculation involving square root would be useful?

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2 Answers 2

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Converting a comment to an answer (at the OP's request), any price function of the form $a(h+w)+b\sqrt{hw}$ with positive coefficients $a$ and $b$ will scale linearly while giving a higher price to a painting of greater area compared to one of lesser area but identical linear size.

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  • $\begingroup$ I found that with dimensions in cm the sweet spot for me was (a = 0.02] and [b = 48] giving a value in $CAD, thanks again - a perfect elegant solution! $\endgroup$
    – Andy
    Commented Oct 8, 2021 at 0:54
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I came up with a good solution using trigonometry.

I regarded the height and width dimensions as if they were right triangle sides. Using these known sides I could calculate the Hypotenuse, which increases inversely to area.

Price = (Height + Width − Hypotenuse) × Factor

p = h + w - √(h² + w²) * f

This still produces a linear price scale, whilst also giving a higher value to a painting with greater area compared to one of a lesser area but identical linear size.

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  • $\begingroup$ I've found a few instances where this doesn;t work out, back to the drawing board... $\endgroup$
    – Andy
    Commented Oct 7, 2021 at 18:05
  • $\begingroup$ Any price function of the form $a(h+w)+b\sqrt{hw}$ with positive coefficients $a$ and $b$ will scale linearly while giving a higher price to a painting of greater area compared to one of lesser area but identical linear size. $\endgroup$ Commented Oct 7, 2021 at 19:03
  • $\begingroup$ How about this example: 5x5 has an area of 25, 3x8 has a smaller area of 24. However my calculations give the 3x8 a higher price, is that wrong? $\endgroup$
    – Andy
    Commented Oct 7, 2021 at 20:29
  • $\begingroup$ I had written your calculation wrong, I understand now why you need a and b, to control the scale right? This is perfect, thank you! Please add this solution as an answer so I can accept it! $\endgroup$
    – Andy
    Commented Oct 7, 2021 at 20:59
  • $\begingroup$ Done. If you find what you consider a "sweet spot" for the ratio of the coefficients $a$ and $b$, I'd be interested. You can ping me with a comment to my answer. $\endgroup$ Commented Oct 7, 2021 at 22:56

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