There are enough hidden homeless people in Britain to fill a city the size of Manchester and they cost the country £1.4bn a year, according to charity Crisis.

The homelessness group has launched a major campaign to improve the lot of the 380,000 people it says are living in hostels, derelict buildings and on friends’ floors.

Crisis said a census was needed to find exact numbers of hidden homeless people and what they need. It kicked off the campaign on Tuesday by installing a mirrored box in London’s Trafalgar Square.

The box is a metaphor for hidden homelessness: those inside the box can see the outside world but people outside cannot see in.

The charity also published a report Hidden Homelessness: Britain’s Invisible City, which gave the costs of hidden homelessness including temporary accommodation costs, benefits and unemployment.

Crisis is also running an email campaign sending messages to 3000 of its supporters to raise awareness of hidden homelessness. The supporters can forward the messages to friends and politicians.

The charity has plans to lobby at the political party conferences in the autumn.

The campaign is part of the charity’s efforts to show the decline of rough sleeping (from 1850 in 1998 to 500 in 2003) does not represent the end of homelessness.

Instead, it says, homeless people are now “hidden” in hostels, squats and bed and breakfasts.

Mark Flannagan, director of communications and campaigning, said hidden homelessness would be a difficult issue to convey to the public: “It’s not as immediately understandable as issues like children or cancer but we are saying it could happen to anyone.”