When it comes to the general roundness of modern coins, there isn’t a single specific answer that can be given for “why round?” The facts do, however, support some reasonable theories. By way of background, nearly all western coinage traditions can be traced back to a common root around 2600 years ago, probably in Lydia (in modern-day Turkey). From there, the idea and the coins themselves spread through trade, cultural exchange, and conquest, and their utility resulted in them being adopted and locally developed across ever-expanding areas.

These first coins were generally of a roundish shape, which appears to be mostly naturally derived from the nature of the material (which was electrum). The first coins were not formed from nice pre-cut planchets, but were instead somewhat irregular, formed on what appear to be stamped or impressed globules of metal. A drop of hot metal will naturally take on a somewhat round shape as it settles, and the act of pressing a design in to it will serve to further that tendancy. If you think of making cookies, where you put a ball of dough on a baking sheet, which becomes a round cookie when baked (or if you press down upon it), that’s sort of what’s happening with early round coins.

The round design was conducive to easy production as well, because the natural tendency of a glob of metal to form a roughly circular shape also means that a shape with corners would require a multi-step process of first manufacturing a consistently thick planchet, cutting out the shapes, and then minting the coin from it, instead of just impressing the design straight in to the initial metal. With time, as weights became better controlled and planchets became standard fare, the round shape continued to be used despite the fact that it might have been easier to use something else. Whether this was out of familiarity, difficulty in production of other shapes, or other reasons isn’t known.

Other traditions of money also lend themselves to a modern round shape for coinage. In some countries and regions, such as Russia, early monetary instruments were in the form of ingots or bars, called Gryvnas. Gryvnas came in various forms in various areas, but were generally of an oblong, bar- or rod-like shape. Lines would be etched in at distances along the bar corresponding to a certain value, and when paying, a sufficient number of lines would be counted, and the bar would be chopped off at that point. This gives us the modern word рубль (ruble), for Russian money – “рубль” comes from the old Russian verb рубить (rubit) meaning “to chop.” Those chopped off pieces of silver would themselves be somewhat round shaped, and analogous to somewhat thick, slab-like coins. Rings and armlets as a form of wealth and exchange (Celtic and Scandinavian traditions have these, among others) also are round in shape, and would lend to an easier acceptance of round coins as an evolution, rather than revolution, in currency. This use of “hrings” can be found in literature of the time, including the epic poem Beowulf.

In addition to bars and ingots, there are actually many other monetary items throughout history that are “coins” in the sense of being commodity metals formed in to items used specifically as money, but which were not round. Square pieces have been minted in many times and places, outside the context of sieges. Small conical and finger-shaped pieces were produced in Sicily around 450 BCE, and stamped… blobs (for lack of a better term) were produced in Georgia in the 13th century. Republican Rome generally used round shapes, but made at least one piece resembling a seashell. Other pieces from around the world have been shaped like arrowheads, dolphins, wedges, balls, knives, and any other pattern that people felt held significance. One of the most well-known is probably the Katanga Cross, an X shaped piece made of copper, weighing up to a couple pounds, used as currency in parts of central Africa as recently as the early 20th century. While round coins do remain the most widespread by far, there are coins today produced (especially collectibles, not meant for circulation) in non-round shapes. In the United States, the strongest force keeping circulating coins round has not been any matter of necessity, but rather lobbyists working on behalf of the vending machine and coin-op industry. The Susan B. Anthony dollar coin, released in 1979, was originally designed to be an undecagon – an 11-sided polygon – to commemorate the 10th anniversary of the Apollo 11 moon landing (the mission patch for which is the reverse design on the coin). The vending industry complained that it would cost them too much to retrofit their machines to take non-round coins, and successfully lobbied congress to force the matter. The coin was rounded off, people had trouble telling it from a quarter, and it subsequently failed to catch on.

Moving on to the question about klippes and siege coins. While it is easier (going way back to the first couple paragraphs of my response) to produce a round coin than a square coin if you aren’t using a planchet of set shape and weight, that is no longer the case when your customers (coin users) expect a bit more consistency and have a lifetime of coin-use behind them. Manufacturing thousands of individual round planchets of consistent size and thickness requires time, special equipment/molds, and a reasonable amount of experience to get things just right. Square planchets, however, can be cut efficiently from a large piece by anybody capable of making a straight line, without leaving any waste behind needing to be re-melted like punching it full of round holes would. In a pinch, in a city without a mint or minting equipment, household silver, jewelry, damaged cannon, church bells, and anything else containing the metal desired can be melted down and poured in to a large plate by any smith or metal worker with a suitable crucible. It could even just be heated and hammered flat on an anvil if it really came down to it. Once the big flat piece exists, mark some lines on it at even distances, slice it in to squares, and hit them with a hammer bearing the emblem of your cause – instant siege currency.

Sources:

Ancient Coin Collecting by Wayne G. Sayles

U.S. Mint & Coinage by Don Taxay

Standard Catalogue of World Coins 1601-1700 edited by George S. Cuhaj

While I don’t know that I can find a link for specific articles, there is excellent material in some back issues of the ANA and ANS journals.