In my area a full size sheet cake serves 108 (2x2x2) and is 18x24
A half sheet is 12x18 and serves 54 (2x2x2)
Industrial sized baking pans are 18"x24", hence "full sheet". Cut that in half and you have 2 pieces 12"x18", thus "half sheets". Cut those in half and you have a total of 4 pieces 9"x12", thus "quarter sheet". For some reason, an extra inch was added to the length of the pans made for home use, so a 9"x13" sized pan is also known as a "quarter sheet".
How many servings out of a rectangular cake depends on the height of the cake.
"Quarter Sheet"
9"x13" x 2" tall (single layer) will give 29 2"x2"x2" sized servings
9"x13" x 4" tall (generally 2 layers w/ filling) will give 58 1"x2"x4" sized servings
9"x12" x 2" tall (single layer) will give 27 2"x2"x2" sized servings
9"x12" x 4" tall (generally 2 layers w/ filling) will give 54 1"x2"x4" sized servings
"Half Sheet"
12"x18" x 2" tall (single layer) will give 54 2"x2"x2" sized servings
12"x18" x 4" tall (generally 2 layers w/ filling) will give 108 1"x2"x4" sized servings
"Full Sheet"
18"x24" x 2" tall (single layer) will give 108 2"x2"x2" sized servings
18"x24" x 4" tall (generally 2 layers w/ filling) will give 216 1"x2"x4" sized servings
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