The acquisition highlights how the sector's reliable cashflow, driven by demand from online shopping, is proving attractive to private equity firms, as well as a spate of takeovers by bigger packaging players.
RPC, Europe's biggest plastic packaging maker, makes a range of products including packaging for beverages, coffee capsules and healthcare products.
Apollo's offer of 782 pence per share represents a premium of 15.6 percent to RPC's closing price on Sept. 7, the last trading day before the offer period began. RPC's shares rose on Wednesday as much as 5.5 percent to 774 pence.
Nicholas Hyett, analyst at Hargreaves Lansdown, said the final offer price was some way behind what analysts had suggested was possible.
Hyett said it was not impossible for a rival bidder to emerge but plastic packaging was far from popular at the moment and with the global economy looking rocky, it might be asking a bit much to expect a rival to pay around 4 billion pounds to keep RPC out of private equity hands.
He also said that with board backing and minimal regulatory concerns, he expected the deal to go through.
The RPC deal is not Apollo's first foray into the plastics and packaging industry. The private equity firm previously owned Berry Plastics, now called Berry Global, a U.S.-based plastics manufacturer and marketer.
Other M&A deals in packaging last year included Australia's Amcor Ltd agreement to buy U.S. firm Bemis Co for $5.25 billion, while DS Smith Plc spent about $2.2 billion to acquire Spanish rival Europac.
The plastics industry is facing tighter regulation in Europe and elsewhere due to environmental concerns.
RPC, which operates in 33 countries and employs about 25,000 people, said in September it was in talks on a possible sale to Apollo and another private equity firm Bain Capital.
The deadlines on the deal for Apollo and Bain were repeatedly extended and Bain Capital ended talks in December.
RPC had said last July that shareholder pressure was preventing it from pursuing some growth opportunities. RPC had been investing heavily to take advantage of China demand as well as produce more recyclable plastics, prompting fears that higher spending would hit cash flow.
Apollo's affiliate will finance the deal through a combination of equity and debt provided by a number of banks.
RPC was advised by Rothschild & Co, Credit Suisse and Evercore. Barclays and Citi acted as lead financial advisers for Apollo.
(Reporting by Sangameswaran S and Arathy S Nair in Bengaluru; Editing by Anil D'Silva and Jane Merriman)
By Sangameswaran S and Arathy S Nair