The earliest Baseball bases were circular objects of all kinds, including plates, which suggested "home plate" itself. So the word dish continues the analogy, even though bases are no longer round.
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Early on, home plate was square like the other bases, with the points of the square aligned with the base lines. However, it was determined that this design made it difficult for the pitcher and the umpire to judge the edges of the plate, and thus the strike zone. To fix this, they just "filled in" the corners facing the pitcher, thereby creating an easier target for the pitcher and umpire to judge the strike zone with.
It is shaped like a house. From the pitcher's viewpoint it looks like a square with a triangle on top of it. This is the same reason why we say the runner from third is going "home."
Because the plate forms the diamond. Theoretically you could have the four bases, go anywhere and use the home plate as the 'home' base and measure out where the other bases belong. Prior to 1900, the home base was shaped like the other 3 bases. It was imbedded in the ground and positioned with an edge pointed at the pitcher's mound. That configuration made the corners just tiny spots. For the 1900 season the owners changed the shape of the home base in an effort to give umpires a better view of the base to be able to call balls and strikes. A base, (first, second, third) is a 12x12 inch square or 144 square inches. The new home base was 216 square inches and shaped like a pentagon with straight lines replacing the edges. Not only did the new shape help the umpires but it gave pitchers a bigger target to throw at.
Home plate is level with the ground to prevent batters (and catchers) from tripping over it after they hit the ball. The other bases are raised to make it easier for runners to stop on the base while running or sliding.
The home base took on its odd shape by design in the early 1900s so that the umpires could have a better judgement of the game, so that they could tell whether a pitch was over the plate or not.
Because it is set-up to frame the strike zone you get four corners for the ball to cross. The point is the last spot a ball could cross on a pitch breaking in and out of the zone