What's London's largest square?
Not a residential quadrangle called Something Square because we've covered that.
I mean actual square as in geometric shape.
I reckon it's this.
It's 16½ miles along each side and it's the largest square you can fit inside the boundary of Greater London. Top left is near Mill Hill East, top right is Hainault Country Park, bottom right is near Knockholt station and bottom left is North Cheam. The centre of the square is in New Cross. The square has an area of about 275 square miles which is 45% of the area of Greater London. It'd be bigger were it not for those annoying indentations in the boundary around Worcester Park (SW) and Woodford (NE).
And you're right, it's not really the largest square you can fit inside Greater London, only the largest where the sides run north/south and east/west. Twist your square and you can do a lot better.
This square is 19 miles along each side. Top left is near Hadley Wood, top right is Emerson Park, bottom right is the hamlet of Horns Green and bottom left is Surbiton. The centre of the square is in Deptford. The square has an area of 357 square miles which is 59% of the area of Greater London. That means the majority of London lies inside this square, which is pretty good for a regular shape inside an irregular shape.
We can do better still with a rectangle.
This rectangle is 20 miles long and 18½ miles wide. It's very nearly the same square as last time but stretched a bit. Top left is near Monken Hadley, bottom left is Hampton Court and the other two corners are as before. The rectangle has an area of 375 square miles which is 62% of the area of Greater London. I was expecting a rectangle would be able to cover an even larger proportion but it turns out London's an awkward shape that's not quite conducive.
London's biggest triangle is even better.
It's longest side is 33 miles long, which isn't far off the longest line you can draw inside London. One end of that longest side is near Harefield Hospital, the other's at junction 28 on the M25. The triangle has an area of 405 square miles which is 67% of the area of Greater London. I find it amazing that you can draw a triangle that encompasses two thirds of the capital, given what a peculiar shape London is, but that lump down south beyond Coulsdon really helps to squeeze in as much land as possible.
I should say at this point that I haven't proven this is London's largest triangle, I've merely jiggled around with straight lines and tried to optimise the area inside. It might well be that a slightly different triangle gives a slightly bigger area, but I reckon ony slightly so 405 square miles is pretty much the largest you can get. Likewise that last rectangle might be beatable, as might that square, but so marginally I still reckon I've got the percentages right to the nearest whole number.
London's biggest circle is easier to be certain about.
Its diameter is just under 21 miles and its centre is reassuringly close to Piccadilly Circus. The circle has an area of 342 square miles which is 56% of the area of Greater London. Again that means we've enclosed more than half of the capital, which I was not expecting to be possible with a circle.
In summary
...circle 56%
...square 59%
...rectangle 62%
...triangle 67%
