Here is how the IRS calculates the penalty.
The IRS expects quarterly payments for the income earned throughout the year. Most people have it withheld from employers/ investments/ payors like SSA. If you owe more than $5,000 you are considered to have substantially underpaid. The IRS will charge a penalty if you owe $1,000 or more so it is worth keeping a check on what you make through the year and getting in those estimated payments or asking to have more withheld. In addition, you must pay interest on the penalty.
See the penalty information here. Some penalties have relief available, see here.
To avoid this in the future, you may want to make estimated payments. You can do that through the program, see How do I make estimated tax payments? or you can pay the IRS online. Don't forget, you may need to pay your state estimated tax as well.
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