Lawn Perimeter Design... Help!

Hi I'm new to this whole gardening thing! We decided to tear out the old lawn and even out the ground a bit. Now it's time to sow lawn seeds and we would appreciate some advice on the design of the boundaries (and probably the rest too!
).



We would like to grow some fruit and veg (courgettes, sweetcorn, tomatoes, strawberries etc), the left bed gets most of
the sunlight so probably there? The string line is currently about
1 meter from the fences. Should we run the lawn all the way to back fence or
maintain a border?
For reference, the fence panels on the left and back
are 1.8 x 1.8m (6x6 ft). We have a couple of small children so the
swing is here for the forseeable future. The garden slopes downwards towards the rear at about 5%.
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Posts
If you're happy strimming the edges the mower misses then go right up to the fence. Alternatively, you could install a mowing strip with sleepers or block paving so you can roll your mower right over it.
Or, you could just have an open border and cut a definitive edge into lawn, if that makes sense.
Not sure if this article will help or not?
https://simplelawncare.co.uk/lawn-design/
That way you'll get a nice sized bit of grass which you can take nearly to the fences at the 'sides' 'bottom' and 'top' but with some good deep planting areas in the 'corners'. You could also make the lawn diamond-shaped instead of square if you prefer straight lines - but you'll get a smaller area of grass.
Both those options will be quite pleasing to the eye and make the garden seem larger.
I wouldn't take a lawn right up to a wall or fence - it's a bu$$er to keep the edges tidy. Long handled lawn shears make light work of edging up to a border or you could install a rim of brick-sized paviours just below the level of the grass. You can mow right over the top of them and you'll hardly ever need to edge the lawn. But it's more difficult if you subsequently want to amend the size / shape of the grass area.
Here are some design ideas with oval lawns.
https://www.google.co.uk/search?q=small+garden+design+oval+lawn&sxsrf=ALeKk03490R5CAXhmHeIPZEO2ZdbAqy27g:1619700156962&tbm=isch&source=iu&ictx=1&fir=QDz7W3ACn4lV7M%2C4l5Jgn1EnH9L5M%2C_&vet=1&usg=AI4_-kQ-sdnyv_GT02ehgNoZEKPS1CShzg&sa=X&ved=2ahUKEwiY3f2jvaPwAhXDxYUKHUNTCy4Q9QF6BAgNEAE#imgrc=QDz7W3ACn4lV7M
You can transfer this technique to the garden to mark out a perfect oval
https://www.google.co.uk/search?q=how+to+make+an+oval+lawn&sxsrf=ALeKk00ZW4wFUt-JlJeW5LajQoVkEPGp2w:1619700259987&source=hp&ei=I6qKYOq0OY2ZlwTjpaTQCg&iflsig=AINFCbYAAAAAYIq4M9wtD3UGumSRWgInbm-TJoCQdgSk&oq=how+to+make+an+oval+lawn&gs_lcp=Cgdnd3Mtd2l6EAMyAggAMgYIABAWEB46BAgjECc6BAgAEEM6CAguEMcBEKMCOgIILjoECC4QQzoECAAQCjoICC4QxwEQrwE6BAguEAo6BQguEMsBOgUIABCRAjoHCAAQhwIQFDoFCAAQywE6CAgAEBYQChAeUMgOWJlpYKFtaAJwAHgAgAHABYgBulKSAQ0wLjMuMi42LjkuNS4xmAEAoAEBqgEHZ3dzLXdpeg&sclient=gws-wiz&ved=0ahUKEwiq0ovVvaPwAhWNzIUKHeMSCaoQ4dUDCAc&uact=5#kpvalbx=_MqqKYLGyIYSWadL1kYAJ83
If the shape turns out to be a near perfect circle, put a stick in the centre and fix a length of rope measured to the radius of the circle - and just walk round marking the perimeter or circumference. If it is to be circle, do make it a perfect one or it will irritate you forever. That said it doesn't have to be a complete circle if (for example) the patio forms a quadrant. Your mind will still see where the circle completes
If it turns out to be more of an ellipse or oval then use one of the methods outlined in the articles found by Obelixx.
The shape you end up with might not be a true geometric shape at all, in which case you can just keep tweaking the hosepipes until you get the right layout.
The one thing I would recommend is that you stick with the principle of big sweeping curves and arcs. Don't be tempted to have lots of little frilly bits eaten out of the edges. It's more difficult to mow and makes the garden look smaller, fussier and bitty. In a small garden you're much better to start off with lovely clean lines.
You can always make it frilly at a later date if that's your thing....
For me 2 S-shaped borders plus an extra bit cut into the lawn at the bottom would be a bit fussy in a garden that size but, yes, it would work if that's what you like.
Try the hosepipe / rope thing - it really will help you to visualise the end result.