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Share-based Payments - Share-based Compensation Expense Recognized (Details) - USD ($) $ in Millions | 12 Months Ended
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Aug. 31, 2022 | Aug. 31, 2021 | Aug. 31, 2020
Share-based Compensation Arrangement by Share-based Payment Award [Line Items] |  |  |
Share-based payment expense | $ 37.4 | $ 32.5 | $ 38.2
Stock Incentive Plan |  |  |
Share-based Compensation Arrangement by Share-based Payment Award [Line Items] |  |  |
Share-based payment expense | 37.4 | 32.5 | 38.2
Restricted stock awards and units | Stock Incentive Plan |  |  |
Share-based Compensation Arrangement by Share-based Payment Award [Line Items] |  |  |
Share-based payment expense | 17.2 | 15.1 | 24.6
Stock options | Stock Incentive Plan |  |  |
Share-based Compensation Arrangement by Share-based Payment Award [Line Items] |  |  |
Share-based payment expense | 8.8 | 9.2 | 4.9
Performance stock units | Stock Incentive Plan |  |  |
Share-based Compensation Arrangement by Share-based Payment Award [Line Items] |  |  |
Share-based payment expense | 9.9 | 6.8 | 7.3
Director stock units | Stock Incentive Plan |  |  |
Share-based Compensation Arrangement by Share-based Payment Award [Line Items] |  |  |
Share-based payment expense | $ 1.5 | $ 1.4 | $ 1.4

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