Stockholders' Equity - Shares of Common Stock Outstanding (Details) - USD ($) $ in Millions	3 Months Ended
	Nov. 30, 2021	Nov. 30, 2020
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning Balance (in shares)	37,615,419	38,030,000
Common stock issued for employee stock plans (in shares)	210,000	117,000
Repurchase of common stock (in shares)	(46,200)	(131,800)
Ending Balance (in shares)	37,772,170	38,008,000
Number of shares repurchased in settlement of employee tax withholding obligations (in shares)	6,747	6,728
Value of shares repurchased in settlement of employee tax withholding obligations	$ 2.9	$ 2.1
Repurchase of common stock from employees
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Repurchase of common stock (in shares)	(7,000)	(7,000)
Share repurchase program
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Repurchase of common stock (in shares)	(46,000)	(132,000)