Paul Millachip was swimming at Port Beach near Perth when he was attacked. The ABC reported that according to witnesses, the man was swimming next to a dinghy when a tiger shark and a great white shark both attacked him. A group of boys who were in a dinghy said they saw the incident.
Only his goggles were found after Saturday's attack, despite a search from both the air and water that was eventually called off yesterday afternoon.
Teenagers witnessed the attack and frantically urged other swimmers to get out of the water.
One witness quoted by The West Australian newspaper said: "It was big, and it looked like a white.
"After the attack we saw it swing out the water with its fin. He was a big one. And then I saw these young guns in a tinny racing up the beach raising the alarm."
Mr Millachip's wife praised the teenagers and said they could have saved other lives, Sky News reports.
"A special mention to those young lads in the boat for what they did in what must have been an absolutely terrifying experience for them, so my heart goes out to them and I thank them for what they did," she said.
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She added: "It came out of the blue. But it did happen, you just have to deal with it."
Mrs Millachip - who is believed to have been in the changing rooms when the attack happened - said she had been in the water earlier, but came out because it was cold.
"We had been going down to the beach two or three times a week," she said.
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"We would go running first and then go swimming; he swam for a kilometre - was due to swim for a kilometre - on Saturday."
Australian media said the beach had reopened on Monday morning but that people were being urged to "remain vigilant".
The country's last fatal shark attack was on 5 Sept in New South Wales.
There have been 17 attacks in Australia this year, two of them fatal, according to the Taronga Conservation Society, an Australian government agency.