It sounds like you starts with the coin covering the hole, does it? you wrote:"Now rotate the coin 30 degrees counter closewise around the matching vertex,then the altitude of the coin starting from the matching vertex will match wih one side of the hole." There are many ways to rotate it around a point. Can you tell me how to rotate it around a line? What do you mean by altitude?
My way is to hold the box (hole) level. Put the coin against the edge of the hole and _|_ to the hole. Also make one of the edge of the coin _|_ to the hole. Then the largest width of the coin going through the hole is the height of the coin, which can be maximum 1. So, the side of the coin can be as much as sqrt(1+(1/2)^2) which is sqrt(5)/2. It is smaller than yours.
My way is to hold the box (hole) level. Put the coin against the edge of the hole and _|_ to the hole. Also make one of the edge of the coin _|_ to the hole. Then the largest width of the coin going through the hole is the height of the coin, which can be maximum 1. So, the side of the coin can be as much as sqrt(1+(1/2)^2) which is sqrt(5)/2. It is smaller than yours.
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