How do I know what's the amount I owe for taxes after getting an extension approved?

[GaryG1]

As part of submitting you extension you should be providing estimated taxes owed. 

TurboTax Easy Extension can help you submit your extension including calculating your tax refund\owed:  https://turbotax.intuit.com/irs-tax-extensions/

You can also use the stand-alone TaxCaster Calculator: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/