Optimizing Picture To Border Ratio With Phi

January 07, 2010 By: erik Category: Art, Geeky, Math 5,993 views

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Frame Geometry DiagramThe other day I was looking at a framed photo in my house and wondering how the area of the border around the picture (called a mat board in the framing industry) compared to the area of the picture itself. Immediately I remembered the master of all aesthetic ratios, the golden ratio, Phi! 1.61803398874989… I thought, “I bet if the ratio of the area of the picture to the border was Phi, it would look good!” So I set off on an algebraic quest to find the answer…

The geometry is simple enough. You’ve got whatever the length and width of your picture is, plus a border around it with a width of x/2.

Frame Geometry Diagram

If we write down the equation for the ratio of the picture area to the border area (the full frame size – the picture’s area), we get this:

Ratio Equation

If we set our ratio to Phi and solve the equation, using the quadratic formula, we get that our doubled border width, x, is:

Border Equation

What does this give us? Now, for any length and width of picture, we can calculate how wide to make the border around it such that the ratio of the area of the picture to the area of the border is Phi…or any number for that matter!

Picture Border Ratio Calculator

Using the calculator below, if you input the length and width of your picture, and the area ratio (picture / border), it will tell you how large to make your border. And below the calculator is a rendering of the values you enter. Note that the default values in the calculator are for a golden rectangle and an area ratio of Phi. Looks pretty nice, doesn’t it? Go ahead, try out other picture sizes!

Picture Length
Picture Width
Area Ratio
Border Width
Or you can use an image from the internet!
Image URL Use Image
 


 
  • http://erik-rasmussen.com/blog Betsy

    I usually just ask the woman who owns the local framing business what she thinks looks good. Sometimes I splurge on a double mat. Woohoo.

  • http://www.hillbillyplease.com/blog/ jane

    Cool.

  • http://www.flickr.com/photos/aparejador/4235582919/ Ray Tibbitts

    I’m not embarrassed to admit that this is really interesting and amazes me, and I’ll bookmark this for future, real world usage.

    • http://www.erik-rasmussen.com/ Erik R.

      Thanks, Ray. That’s a huge compliment.

  • http://www.photolp.com Laura

    This is exactly what I’ve been looking for! But the calculator doesn’t seem to be working. Any chance of fixing it?
    :)
    Thanks!

    • http://www.erik-rasmussen.com/ Erik R.

      Thanks for the heads-up, Laura. I recently removed a javascript dependency from my blog and forgot about this post. The calculator should work now. I hope you find it useful!

  • Shehan Khan

    Excellent work…. Kudos… Need more people to do stuffs like this…. Thanks a zillion…

  • Morris

    I would love to see the math for inputting a frame size and then having it tell me what the border and image size would be to fit.

  • Bert

    Would this just be the mat or inlcude the frame plus mat?  If it doesn’t include the frame, then how would you get a good looking frame width.  Would you then want to take the mat plus picture area to frame area to get the frame width?

  • phil

    Interesting approach eric.
    Can i suggest a different algorithm based on the famous ‘thirds principle’ in photography. “That the picture width and length are 2/3rds the overall matted width and height.”
    For example, assume you have a photograph 18×12 inches. Then the overall matt dimensions become 27x18inches.respectively.
    Note that the matting width for length and height are different.
    This gives a subtle effect of making the photograph appear larger.
    The area ration with this algorithm is always 2.25, substantially larger than Phi. But I like it more than the Phi dimensions

  • SeraphX

    Exactly what I was looking for! I was guessing I’d use a 3.25 inch wide frame and your calculator answered 3.27. A confirmation of my golden eyeball. Thanks!

  • Claudio Rosso

    Can you help me in the inverse problem ?
    I have a frame and I need to calculate the size of the picture !
    Thank you