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Height, breadth, and depth: the British Museum possesses these dimensions in abundance, both in the solid angles of the neo-Classical exterior and the whimsical flourishes of the Great Court. Its treasures extend back along the dimension of time for millennia. But, as I discovered in a single day's immersion in this English miracle, you need an extra dimension to get the most out of the collection: guides Helen, Keith or Carolyn, each of whom add the essential extra aspect of context.

One reason London is so popular with both British and overseas tourists is the policy of free admission to the nation's great museums and galleries. The British Museum tops the cultural pyramid, welcoming strangers and scholars to share in the historic riches it has accrued (sometimes controversially) since it was founded in 1753.

Yet many of the millions of visitors to this people's palace are unaware of another gift: the daily programme of "eyeOpener" tours, led by expert volunteers, who offer insights for the benefit of the curious and culturally challenged.

The Marxist view of history that prevailed at my school in Crawley in the Seventies places me firmly in the latter camp: I had been led to conclude that nothing much of interest had happened before the Agrarian Revolution. Helen, Keith and Carolyn put me right, each of them celebrating a specialism that shows the sheer depth and diversity of the British Museum.

"Art of the Middle East", with Helen, opened my eyes to the wealth of imagination and expertise that prevailed from Spain and north Africa to Asia while north-western Europeans shivered and stumbled through the Dark Ages.

In the 45 minutes that each eyeOpener tour lasts, you can learn an astonishing amount: from the spread of Islam from its heartland in Arabia, via the techniques used to glaze tiles, to the themes that recur in sacred and secular art of the Middle East. And all illustrated with examples whose exquisite artistry and intricate detail reveal much about the world whence they came.

Later, with his "Early Medieval Europe" tour, Keith pointed out that life wasn't entirely dreary in England in the 6th and 7th centuries. He celebrated the remarkable artefacts found by an amateur archaeologist at Sutton Hoo in Suffolk in 1939, which revealed a ship burial and a trove of objects from weapons and helmets to jewellery and ornaments. The society that mourned their king was in no sense isolated from the rest of Europe – indeed, Keith traced the trading connections with the Mediterranean, and led on to the arrival of the Vikings (or, "pirates", as he succinctly described them).

Keith's tour ended with the recommendation to check out the Lewis Chessmen, pieces sculpted from walrus ivory and whales' teeth with skill and subtlety to create an ensemble that could be 21st-century rather than 12th-century. Back to the future, indeed.

Rating a "premier league" of objects is a tricky business for any collection; to its credit, the British Museum offers a self-guided tour that allows the time-pressed to make the most of 60 minutes in Bloomsbury. The Lewis Chessmen make the grade, as do the Assyrian reliefs that Carolyn guided me through on her eyeOpener on these gypsum friezes, nearly 3,000 years old. They occupy a long, narrow gallery, and many of the people passing through were evidently blissfully unaware of the stories and significance of such antiquity. Carolyn explained the regal significance, unravelled the lion hunt depicted, and pointed out artistic twists such as the detail on the be-sandalled toes.

"Visit the British Museum – see the world". It's a catchy slogan, and a controversial one, too; the opening of the new Acropolis Museum in Athens today draws yet more attention to the Parthenon Friezes (the Elgin Marbles) and the rights and wrongs of removal and restoration. It is beyond dispute, though, that the British Museum celebrates its collection with all-comers, frequently and freely.

Traveller's Guide

The British Museum is at Great Russell Street, London WC1B 3DG (020-7323 8000; britishmuseum.org). It opens 10am-5.30pm daily, with late opening of some galleries to 8.30pm on Thursdays and Fridays. The Great Court is open 9am-6pm from Sunday to Wednesday, 9am-11pm from Thursday to Saturday. Admission to the Great Court and main collection is free, with charges made for some special exhibitions.

The daily programme of eyeOpener tours begins each day with Japan at 11am, and concludes with Assyrian Reliefs at 3.45pm. In addition to these free tours, there are "Highlights" guided tours led by Blue Badge guides, for which a charge is made. In addition, there are special gallery talks with guest speakers or curators; details of these are available online.

'A different magnitude': The British Library

The largest public building constructed in the UK this century is Heathrow Terminal 5. Yet you may be surprised to learn that the 20th-century title goes to an institution an hour away on the Piccadilly Line, on the northern edge of central London.

Graduates of universities founded in the Sixties will feel strangely comfortable as they approach the British Library: Colin St John Wilson's brickwork (10 million of them, manufactured to match neighbouring St Pancras station) and the awkward angles of this massive structure resonate with the campuses of Sussex and Warwick. But the business of providing space for endless shelves and studious readers proved a project of an altogether different magnitude in terms of both time and money.

Ten years after the British Library opened, I took advantage of the free "backstage tour" that gives an insight into the audacious scale of the project – which helps you to understand why it took 25 years and cost a fortune.

The tour explains the idea of the library, as a legal deposit for every book published in the British Isles, and its role in providing access for researchers. You are taken behind the scenes to see the reading rooms from a viewing gallery, where you can almost feel the brain power being expended on making connections and elaborations from the printed word. My brain felt a little strained by all the statistics about the miles of shelves and time taken to fetch a book by a Terminal 5-style baggage-handling system.

At the end of the tour you are left happily at the entrance to the Sir John Ritblat Gallery, home to documents such as the Magna Carta and the Lindisfarne Gospels.

But I reckon the highlights are some of the most influential scraps of handwriting in history: the last despatch from Antarctica by Captain Scott in 1912, and the scruffy lyrics of "I want to hold your hand", by John Lennon and Paul McCartney.

SC

Free gallery tours of the British Library take place on Mondays, Wednesdays and Fridays at 11am. Call 0870-444 1500 or see bl.uk for more information