Measure the length and width of the air vent. Measure the distance from the top of the vent to the ceiling and the distance from the bottom of the vent to the floor. Measure the horizontal distance from the adjacent piece of drywall to the top and bottom of the air vent.
Transfer your measurements to the fresh piece of drywall. Draw the vent's outline on the drywall with a pencil. Use a carpenter's square as a guide to ensure that you draw perfectly straight lines.
Insert a utility knife into the corner of the traced vent on your drywall square. Move the utility knife around the traced area, following the lines.
Push the outlined vent out of the drywall.
Lift the drywall up to the wall. Fit the drywall's hole around the vent. If you measured and cut the hole correctly, it will fit snugly around the vent.
Apply joint compound to edges of each piece of drywall and any other area with ragged edges. Smooth out the ragged edges with a drywall knife. Let the joint compound dry.
Apply paper drywall tape to the edges where you spread the joint compound. Apply another layer of joint compound over the tape. Smooth out the joint compound over the tape and allow it to dry completely.