The formula below returns the day of the year for a date in cell A1:
=A1-DATE(YEAR(A1),1,0)
Note: Excel automatically formats the cell as a date, so change the number format to another option (like General).
To calculate the number of days remaining in the year (assuming that the date is in cell A1), use the following formula:
=DATE(YEAR(A1),12,31)-A1
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