Stockholders' Equity - Shares of Common Stock Outstanding (Details) - USD ($) $ in Millions	3 Months Ended		9 Months Ended
	May 31, 2022	May 31, 2021	May 31, 2022	May 31, 2021
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Balance, beginning of period (in shares)			37,615,419	38,030,000
Common stock issued for employee stock plans (in shares)			409,000	280,000
Repurchase of common stock from employees (in shares)			(7,000)	(7,000)
Repurchase of common stock (in shares)	0	(178,100)	(46,200)	(531,859)
Balance, end of period (in shares)	37,971,366	37,771,000	37,971,366	37,771,000
Value of shares repurchased in settlement of employee tax withholding obligations			$ 3.3	$ 2.4