Math to scale images to the same pixel surface area proportionally

[ultrus]

I know how to scale images down to a max width/height proportionally, but have an interesting logo challenge.

 

Some logos are tall. Some Logos are wide. Some logos are square. If I were to scale down a tall, wide, and square logo to a max size of 500x500 pixels, graphic gurus would say that the square logo looks huge compared to the others. I would then need to manually resize them all manually to look 'good'.

 

However, if I scaled down each logo to a max surface area of say 5000 total pixels (or something just under that number), that would work better.

 

Would I need to create a loop to calculate the image at smaller increment sizes to figure out what size my new image would be, or is there an easier math solution for that?

 

Any thoughts? I don't need specific php examples. Just trying to get the concept wrapped around my head this Friday afternoon.

 

Thanks in advance!