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Income Taxes - Narrative (Details) - USD ($)	Aug. 31, 2023	Aug. 31, 2022	Aug. 31, 2021	Aug. 31, 2020
Income Tax Contingency [Line Items]
Undistributed earnings and original investments in foreign subsidiaries	$ 255,800,000
Deferred income tax liability for certain foreign withholding taxes and U.S. state taxes	700,000
Unrecognized tax benefits	20,100,000	$ 19,500,000	$ 17,700,000	$ 17,200,000
Unrecognized tax benefits that if recognized would affect annual effective tax rate	20,100,000	18,800,000
Total accrued interest	3,300,000	$ 2,100,000	$ 2,000,000
Accruals related to income tax penalties	0
Internal Revenue Service (IRS)
Income Tax Contingency [Line Items]
Federal tax credit carryforwards, subject to expiration	8,300,000
Tax credit carryforward subject to a full valuation allowance	7,600,000
Operating loss carryforwards	14,400,000
State and Local Jurisdiction
Income Tax Contingency [Line Items]
Federal tax credit carryforwards, subject to expiration	900,000
Operating loss carryforwards	47,100,000
Foreign Tax Authority
Income Tax Contingency [Line Items]
Operating loss carryforwards	$ 7,800,000