A System for Sizing Chess Pieces and Boards (long)

 @loubalch You are truly a gentleman and a scholar!! Amazing work!

Another wrinkle:

FIDE recommends that the king's diameter to height ratio is from .4 to .5. Given that, and the assumption that the king's diameter to square size ratio is from .7 to .8. The OP has a wider range - but I am lazier. Using the FIDE recommended range, I'll say the ideal diameter to height ratio is .45 (mean of the range). In the same spirit, the ideal diameter to square ratio is .75.

So, if the king's height is 4" the acceptable FIDE king diameter range is from 1.6" to 2.0". The ideal diameter base is 1.8" which makes the ideal square is 2.4". Actually the diameter range from 1.6 to 2.0 can be computed:

$ python khdratio.py -k 4
diameter: 1.600 - 2.000
ideals: diameter: 1.800 | square: 2.400

dia | minsq | ideal | maxsq
1.600 | 2.000 | 2.133 | 2.286
1.650 | 2.062 | 2.200 | 2.357
1.700 | 2.125 | 2.267 | 2.429
1.750 | 2.188 | 2.333 | 2.500
1.800 | 2.250 | 2.400 | 2.571
1.850 | 2.313 | 2.467 | 2.643
1.900 | 2.375 | 2.533 | 2.714
1.950 | 2.438 | 2.600 | 2.786
2.000 | 2.500 | 2.667 | 2.857

That ideal diameter of 1.8 is very consistent with the 4" Jaques sets.

Here is a table I put together awhile back that provides the minimum and maximum acceptable King dimensions based on USCF regulations. Oh, and don't forget, if the pawns aren't properly matched, it doesn't matter how well the king fits.

well that took a while...

So I played at a tournament just this last weekend and a guy told me that if 4 pawns don't fit inside the square then it doesn't fit the USCF guidelines. All the pieces including the king fit the guidelines its only that I couldn't fit 4 pawns in one square. I'm confused.

That person was wrong.

When someone says something like to you ask them to show you in the official rules. 

The 4 pawns thing is a guideline some people use. 

Some say they should completely fit inside, some say they can overhang a little bit.

Just find the official USCF rules, print them off, and throw them in your tournament bag. 

Thank you Rsava for responding. I've been scouring the internet trying to find the specific guidelines for USCF equipment standards but it seems that you have to purchase their rule book. I heard people say that you can use two pawns placed diagonally in one square to fit the USCF guidelines but no one has confirmed this too me and I have yet to buy the rule book. Thanks again for your response.

Not official but it references the official rules:

https://www.chessusa.com/chess-pieces-size.html

Down at the bottom of this page:

https://www.wholesalechess.com/chess-sets-buying-guide

And finally on this page Rafael (owner of the site) quotes the rulebook:

http://blog.chesshouse.com/how-to-select-the-right-size-chessboard-for-pieces/

Thank you Rsava for you input and taking the time to find these reference sites. I appreciate your input and I find them helpful but I am still left with the question of the official rule on the size of the pawns as only the chessusa site speaks briefly about this subject without giving a specific ruling on official USCF rule. I fear I will be forced to buy the rulebook which is a downer. I believe when you pay for your annual USCF membership you should be given at least a copy of standard equipment guidelines and I don't see why it's so hard to find these in pdf format in modern day. My quest continues... Thanks again for all the input.

Ok so I purchased the ebook USCF Official Rules of Chess 6th edition by Tim Just and no where in it does it say 4 pawns have to fit into a square. The only real specifications in the book are the kings height should be at least 3 3/8 to 4 1/2 inches tall and have a base diameter of 40-50% of the height and that the kings finial be no more than 20% of the total height. Boards for standard sets should have squares of approximately 2 to 2½ inches (5.08 to 6.35 cm). The guideline for determining the proper square size for a Staunton chess set is that the king should occupy around 78 percent of the square. (Dividing the base diameter of the king by 0.78 will yield the proper square size). An acceptable square size may be up to ⅛ inch larger than this number, but not smaller. This is really the only clear set rules in the USCF rule book for the 6th edition for all those who may be wondering.

The initial post in this thread is very interesting. One can quibble about this, though: if one is choosing a chess board to go with an existing set of chess pieces, there's no point to specifying an ideal ratio of King base diameter to Pawn base diameter, as that's not one of the things that one is able to choose.

Hi, as the OP, I drew up my calculations because I had a traditional Staunton style set where the diameter of the pawns was half the size of the square (or 4 pawns to a square). When I selected a board to fit the king, based on USCF and FIDE guidelines, I noticed that the pawns looked decidedly undersized, as if they belonged to a smaller chess set. It was then that I realized that the problem went beyond sizing the king to a chess board. Unless the pawns are scaled to fit the king, achieving a balanced look would always be a compromise. Fit the king to the board and the pawns are swimming in their squares. Fit the pawns to the square and the king looks like a corpulent whale bursting at the seams.

In essence, the calculations were created to help in selecting well balanced chess sets where both the king and pawns are properly scaled to the right size board.

As an example, I've taken the following Staunton style set and matched the king to the appropriate size board with the king at about 75-78% of the sqaure. The first two pics show this set with classically sized pawns at four pawns to the square. The second two pictures are based on a scheme where the pawns are sized so that two pawns will just fit when place diagonally inside the square.

There's no right or wrong way, both are acceptable standards for tournament play. It's just, aesthetically, I much prefer the larger pawns.

My point being, pawns are chess pieces too. And since the make up half of all the pieces on the board, I think their dimensions should be taken into consideration when designing chess sets, and when sizing chess pieces and boards.

If you look around, there are plenty of sets with larger pawns. Below is a plastic Cavalier set on a 2" board. The king is just under 79% of the square and the pawns are 56%.

I did realize that this was your goal, and that choosing a chess set with the right design would be involved. My criticism was simply that I felt that your initial post could be confusing people, as they would assume it was only about selecting the right board.

Since then, I've taken a close look at the USCF rules on board sizes. Their rule is that the King diameter must be no more than 78% of the square size, and that the square not be more than 1/8" larger than the minimum size square that determines.

Well, this explains why they sell chessboards with 2", 2 1/8", 2 1/4", 2 3/8" and 2 1/2" squares. For any given tournament-size chess set, one, and only one, of those sizes would be permissible to use with that chess set!

So after the size of the King, no other variables count for the purpose of satisfying their criterion.

I agree that this is wrong-headed.

But it's not clear to me that the cure is to consider how the Pawns fit in the squares. The Pawn, after all, is the smallest piece. So, if there's an ideal proportion of a piece to a square, then, if the set is properly proportioned to the board, the Pawn will be smaller than that proportion. And the King will be larger.

So if I had been setting the standards for boards and pieces in tournaments, I would be inclined to construct them as follows:

Obviously, the King can't be allowed to be bigger than the square. So one could set, as an absolute maximum, that the base diameter of the King not be more than 90% of the side of the square.

And maybe also a rule is needed to prevent the Pawns from being excessively small. So it could be required that the base diameter of the Pawn not be less than 40% of the side of the square.

The idea would be that those would be very loose limits that in themselves don't guarantee that the chessboard is a good fit for the pieces.

Instead, the primary criterion would be based on comparing the Knight, the Bishop, and the Rook to the board. Because they represent the 'regular' piece size, and so that is what should be best harmonized to the board. A possible standard might be that the average of the base diameters of those three pieces is to be between 64.5% and 69% of the side of a square - put the three pieces next to each other, and if their combined widths just about equal two squares, the criterion is met.

That doesn't prevent it from being noted as also a good thing if the sizes of the pieces don't differ in too extreme a fashion, but it starts, in my opinion, from what is most important, and allows for maximum stylistic freedom in the design of chess sets - which can then be re-examined in the light of results.

Here's a simply formula that will assure that the kings and pawns are properly scaled to a board (and to each other). I call it the 765 rule. Take any size board. Multiple the square size by 76.5% (.765), this gives you the diameter of the king. Now, take the king diameter and multiply that by 76.5% (.765), this gives you the diameter of the pawn.

Example: for a standard 2.25" chess board,

Sq (2.25") x .765 = Kd (1.72") x .765 = Pd (1.32")

This will give you a pawn whose diameter is 58.6% percent of the square. A diameter where two pawns, when placed diagonally, will fit exactly inside the square.

This is a really old thread, but someone was asking about the 4 pawns to the square rule.  USCF rules say "about 4 pawns" so it's just a guideline.  Not so for the International Chess Federation.  They require 4 pawns to fit in a square.
From their Rule book:

"Size of the square and the board

The side of the square should measure 5 to 6 cm. Referring to 2.2, the side of a square should be at least twice the diameter of a pawn’s base (it means four paws on one square). "

Interesting that the sets they use for GM tournaments do NOT adhere to their own specifications. In fact, their own FIDE Championship Chess Set, which is always mated with a DGT e-board has a king that's 70.9% the size of the square and the pawns are 54.5%, both dimensions are outside of their own rules/regs/guidelines, whatever you call them. So I guess it's, "Do as I say, not as I do."

Sets with kings with a diameter of 68-70% of a square still looks very good to me. This also seems standard in European Tournament play. In Netherlands you don't have to bring your own chess set. At the Wijk aan Zee Tata Steel chess tournament all amateur sets had 68% kings with four pawns on the square. 78% would look way too crowded to me.

My Dubrovnik chess set from Chessbazaar with 32mm diameter pawns and 42mm diameter king looks perfect to me on 60mm squares. While most people have these on 57mm squares.

In the end it's all personal. 

I am reminded of the time that Anand and his challenger did an a la carte selection of the DGT pieces to be used for the world championship match. If only I can find that article on Chessbase...

Found it!

https://en.chessbase.com/post/world-championship-in-bonn-checking-the-playing-venue

Sadly the pictures are gone.