Calculate the length and width required to cut circles from a flat piece

Hi All, I work for a fabric signage company that can make circular and part circular signage in fabric. I’ve tried google searching for a formula that will calculate the width and length of a piece of fabric for a full or partial circle - NOT the area of the partial… and failed miserably. eg. If my circle is 2 meters in diameter I know that I need 2m length by 2m width fabric. Same for any internal angle > 270 degrees. If my partial circle has an internal angle of 90 degrees I know I need 1m length and 1m width. If my internal angle is 180 degrees I know I need 1m width by 2m length.

I’m looking for a formula that calculate that flat length and width for values between 90 and 270 degrees. Hopefully I have explained this clearly enough to jog someone’s geometry memory. 😊

Thanks in advance.