Average UK house prices are up again – by £3,000 in just one month! Yay, huzzah, let joy be unconfined, woo.

Or, to put it another way, we're screwed.

Much of the problem is that (spoilers) we don't have enough houses; and much of the reason for that can be traced to the fact that we don't really have anywhere to put them. A research note published last week by Savill's researcher Neal Hudson makes the point.

In all, Hudson says, drawing on government figures, just 11 per cent of England is developed. Which isn't very much, and implies that the whole “this country is full” narrative is nonsense.

But that figure understates quite how much land is free to build on: much of England, for one reason or another, is untouchable. In all 13 per cent of the country is classified as “green belt”. National Parks, Areas of Outstanding Natural Beauty (AONBs), and Sites of Special Scientific Interest between them take out 29 per cent.

In all – the numbers don't quite add up, because the two categories overlap – around 40 per cent of England is out of bounds to housebuilders and pretty much anyone else with a JCB.

Things get worse when you look at the area where housing pressure is greatest. Hudson's note also includes a map of the south east of England. Blue is green belt (obviously), and red is land covered by other restrictions – AONBs, National Parks, flood plains, et al. Dark grey is land that’s already built up.

Image: Neal Hudson/Savills.

Which leaves white. So that’s where we can actually build, right?

Well, er, maybe not actually. To explain why, it's worth zooming in on London. We've labelled a few of the larger white areas.

Image: Neal Hudson/Savills.

Now some of these areas – the Wandle Valley, the various ex-industrial wastelands that make up “London Riverside” – are already seen as development areas.

But others have got to be off limits. “Building housing in Heathrow Airport” is surely a non-starter to anyone except Boris Johnson. Meanwhile Hampstead Heath, Hackney Marsh, and the “green chain” of parks in south east London may not technically be green belt; but in any sane world, surely they should be treated as sacrosanct anyway, right?

On closer inspection*, in fact, many of these places are – they come under the London-specific designation of "metropolitan open land". For some reason this seems to have been missed from the map.

(Full disclosure: I'm not actually sure what's going on with the gap in the landscape around the A40 just north of RAF Northolt, and after googling for 10 minutes, gave up trying to work it out. Get it touch if you fancy plugging this gap in my knowledge.)

So – there's hardly any land in London that's free for development, and large chunks of that which there is isn't really free for development at all.

Two lessons present themselves from all this.

1. A land use policy that makes huge swathes of golf courses clinging to the M25 untouchable, but fails to protect Hackney Marshes, is just possibly not fit for purpose.

Seriously, look at the state of this.

Annotated extract from Google's map of the Ilford-Romford area of east London.

2. If you can't build outwards, the only way is up.

That's because, with almost no spare land that is both a) empty and b) untouched by land-use restrictions, the only option left to house the 100,000 or so people who arrive in London each year is to redevelop land that's already built up. "Brownfield" will help but, according to estimates we've covered before, will get you less than halfway to meeting London's need.

So, to plug the gap, we either have to knock things down and rebuild them at higher density (this tends to focus on council estates, and go down brilliantly with the people who live on them); or take smaller patches of land, but build them up much, much higher than they were before.

If you ever wondered why London's skyline is becoming more and more crowded with skyscrapers, this is why.

Jonn Elledge is the editor of CityMetric. He tweets, far too much, as @jonnelledge.

*NOTE: This paragraph wasn't in the original article. It was added, in response to comments on social media, on the morning of 19 August. Never let it be said we are unresponsive to reader comments, except when we don't want to be.