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Share-based Payments - Narrative (Details) - USD ($)	12 Months Ended
	Aug. 31, 2024	Aug. 31, 2023	Aug. 31, 2022	Jan. 31, 2022	Dec. 31, 2021	Aug. 31, 2021
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Share-based payment expense	$ 46,600,000	$ 42,000,000.0	$ 37,400,000
Omnibus Stock Compensation Incentive and Directors’ Equity Plans
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Number of shares authorized for issuance (in shares)				3,600,000	2,700,000
Number of shares available for grant (in shares)	700,000	1,000,000.0	1,100,000
Stock options outstanding (in shares)	900,000	1,000,000.0	1,100,000			1,200,000
Share-based payment expense	$ 46,600,000	$ 42,000,000.0	$ 37,400,000
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Restricted stock awards and units
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Shares outstanding of restricted stock (in shares)	300,000	300,000	300,000			400,000
Unrecognized compensation cost	$ 30,100,000
Unrecognized compensation cost period of recognition (years)	1 year 3 months 18 days
Fair value of shares vested	$ 22,900,000	$ 19,900,000	$ 16,400,000
Awards that vested in the period (in shares)	200,000	100,000	100,000
Share-based payment expense	$ 23,900,000	$ 19,600,000	$ 17,200,000
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Restricted stock awards and units | Grants awarded beginning 2022
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Share-based compensation vesting period (years)	3 years
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Restricted stock awards and units | Grants awarded prior to 2022
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Share-based compensation vesting period (years)	4 years
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Stock options
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Unrecognized compensation cost	$ 700,000
Stock options outstanding (in shares)	900,000
Award expiration term	10 years
Intrinsic value of options exercised	$ 6,600,000	500,000	14,000,000
Total intrinsic value of stock options outstanding	116,500,000
Total intrinsic value of stock options expected to vest	10,100,000
Total intrinsic value of stock options exercisable	106,400,000
Share-based payment expense	$ 2,600,000	$ 5,700,000	$ 8,800,000
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Stock options | Market Options
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Share-based compensation vesting period (years)	4 years
Stock options outstanding (in shares)	300,000
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Performance stock units
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Shares outstanding of restricted stock (in shares)	300,000	200,000	200,000			200,000
Share-based compensation vesting period (years)	3 years	3 years
Unrecognized compensation cost	$ 12,600,000
Unrecognized compensation cost period of recognition (years)	1 year 4 months 24 days
Fair value of shares vested	$ 5,000,000.0	$ 11,500,000	$ 0
Awards that vested in the period (in shares)	100,000	100,000
Performance stock units outstanding	300,000
Share-based payment expense	$ 18,600,000	$ 15,200,000	9,900,000
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Employee Deferred Stock Units
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Shares vested (in shares)	4,000
Share-based payment expense	$ 0	$ 0	$ 0
Omnibus Stock Compensation Incentive and Directors’ Equity Plans | Restricted Stock Units (RSUs)
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Shares vested (in shares)	38,000
Employee Stock Purchase Plan
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Number of shares authorized for issuance (in shares)	1,500,000
Number of shares available for grant (in shares)	1,000,000.0
Employee discount on purchases of common stock	5.00%

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Period from grant date that an equity-based award expires, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.

+ References

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- Definition

Amount of difference between fair value of the underlying shares reserved for issuance and exercise price of vested portions of options outstanding and currently exercisable.

+ References

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