Leases - Schedule of Other Information Related to Operating Lease (Details) - USD ($) $ in Millions	3 Months Ended
	Nov. 30, 2021	Nov. 30, 2020	Aug. 31, 2021
Leases [Abstract]
Operating lease cost	$ 10.5	$ 10.7
Variable lease cost	$ 2.9	3.4
Weighted average remaining lease term (in years)	9 years 2 months 12 days		9 years 4 months 24 days
Weighted average discount rate (incremental borrowing rate)	4.30%		4.30%
Cash paid for amounts included in the measurement of lease liabilities	$ 11.1	9.7
Lease ROU assets obtained in exchange for lease liabilities	$ 1.4	$ 1.1