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Property, Plant, and Equipment - Schedule of Property, Plant, and Equipment (Details) $ in Millions	3 Months Ended	9 Months Ended
	Feb. 28, 2022 USD ($) building	May 31, 2022 USD ($)	May 31, 2021 USD ($)	Aug. 31, 2021 USD ($)
Property, Plant and Equipment [Line Items]
Total property, plant, and equipment, at cost		$ 877.7		$ 845.3
Less: Accumulated depreciation and amortization		(608.5)		(576.2)
Property, plant, and equipment, net		269.2		269.1
Gain on sale of building		2.3	$ 0.0
Sold
Property, Plant and Equipment [Line Items]
Building classified as held for sale | building	1
Property, plant and equipment held for sale	$ 6.6
Gain on sale of building	$ 2.3
Land
Property, Plant and Equipment [Line Items]
Total property, plant, and equipment, at cost		22.3		22.4
Buildings and leasehold improvements
Property, Plant and Equipment [Line Items]
Total property, plant, and equipment, at cost		201.5		198.0
Machinery and equipment
Property, Plant and Equipment [Line Items]
Total property, plant, and equipment, at cost		$ 653.9		$ 624.9

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