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Acquisitions and Divestitures (Details) $ in Millions	1 Months Ended	12 Months Ended
Acquisitions and Divestitures (Details) $ in Millions	Nov. 30, 2022 USD ($)	Aug. 31, 2024 USD ($) acquisition	Aug. 31, 2023 USD ($)	Aug. 31, 2022 USD ($) acquisition	Jun. 01, 2024 USD ($)
Business Acquisition [Line Items]
Number of acquisitions \| acquisition		0		0
Cash outflows for working capital settlements		$ 0.0	$ 35.5	$ 12.9
Goodwill		1,098.7	1,097.9	1,084.3	$ 1,100.0
Loss on sale of business		0.0	$ 11.2	$ 0.0
Disposal Group, Not Discontinued Operation, Gain (Loss) on Disposal, Statement of Income or Comprehensive Income [Extensible Enumeration]			Other Nonoperating Income (Expense)
Disposal Group, Not Discontinued Operations \| Sunoptics Business
Business Acquisition [Line Items]
Assets held for sale, total	$ 15.1
Loss on sale of business	$ 11.2
Fiscal 2023 Acquisitions
Business Acquisition [Line Items]
Goodwill		15.0
Identifiable intangible assets		$ 18.0
Weighted average amortization period (in years)		15 years
Intelligent Spaces Group Segment
Business Acquisition [Line Items]
Goodwill		$ 15.0