Income Taxes - Reconciliation of Effective Tax Rate (Details) - USD ($) $ in Millions	12 Months Ended
	Aug. 31, 2020	Aug. 31, 2019	Aug. 31, 2018
Income Tax Disclosure [Abstract]
Tax at U.S. Federal statutory tax rate	21.00%	21.00%	25.70%
State and local taxes, net of U.S. federal income tax benefit	3.10%	4.00%	2.90%
Foreign income at other than U.S. rates	(1.40%)	(1.40%)	(3.20%)
Foreign derived intangible income ("FDII") deduction	(1.80%)	(1.70%)	0.00%
Domestic production activities deduction	0.00%	0.00%	(1.60%)
Income tax benefits from R&D tax credits	(3.80%)	(3.50%)	(3.70%)
Share-based payments	(3.70%)	(3.20%)	(2.70%)
One-time transition tax from TCJA	0.00%	(0.40%)	6.60%
Other, net	(0.70%)	1.60%	0.10%
Effective tax rate	12.70%	16.40%	24.10%
One-time transition tax expense (benefit)		$ (3.4)	$ (23.2)