Please make a drawing of that. You know exactely it doesnt work like that.

Assuming you do it perfectly, and the surface is perfectly spherical, it does work like that.

The angle sum of the triangle is given by:

A=pi + A*K, where K is the curvature, for a sphere equal to 1/r^2.

So for Earth's ~6371000 m radius, the curvature is ~2.4*10

^{-14} m

^{-2}.

So if you covered an area of 100 m

^{2}, the spherical excess would be ~2.5*10

^{-12} radians, or ~5.1*10

^{-7} arcseconds.

That means if you took a bar along that path, it would have appeared to rotate by ~5.1*10

^{-7} arc seconds, compared to how it started.

If you did it 1 billion times, it would have appeared to rotate by ~0.14 degrees.

If you did it 1 trillion times, it would have appeared to rotate by ~140 degrees.

In practice, it doesn't work like that.

Each time you turn the corner you likely introduce a small error in the angle, and the surface is likely not a perfect shear. If it is irregular then instead of just using the Gaussian curvature of a sphere, you need to integrate over the area, including the Gaussian curvature of all of it.

For example, with the simple extreme case of an almost perfect cube (which is topologically equivalent to a sphere) with an extremely large Gaussian curvature at the corners (due to the edges and corners being rounded ever so slightly to avoid having sharp edges with infinite curvature) and a Gaussian curvature of 0 elsewhere, then for most of the surface it has an excess of 0 degrees, but each corner you include in the shape gives an excess of 90 degrees.

Therefore we can conclude one of 2 things

His patio is not as flat as he says (Earth is curved)

His patio is flat like he says (Earth is flat)

And by that lack of reasoning we can conclude that any round surface shows Earth is round.

In order to claim otherwise you need to explain a flat patio could not be constructed on a round Eath.

If the Earth is curved then the end of one end of the patio would not line up with the end of the other end. Take your laser and hold it 1cm off the ground and put a target on the other end of the patio 1cm off the ground. If the Earth is curved, the laser will not hit the target but shoot off into space

Comprehend?

That is if it is level, not flat.

If you construct it flat, then it does.

Flat and level are different.

You can construct a flat object which is not level.

Comprehend?

If you would like an example, consider LIGO.

An object which was constructed to be "flat", in the sense that the light paths are straight, rather than following the curvature of Earth's surface.

This could have been constructed in one of 2 ways and achieve the same result.

The simplest and fastest (due to the large size) is to account for the curvature and terrain of Earth so construction can begin everywhere. Each location can know how far down to dig and go much faster than the latter method. Then precision alignment only occurs at the final stage.

The more time consuming is to start building in one location, and then align everything to it. This works regardless of the shape of Earth, it just means if you didn't dig down far enough at the first spot, you need to dig further down and scrap most of the effort put in.

Imagine paving the surface of that entire red line (outer shell of the Earth) so the whole thing is a patio. Is the patio 'flat'?

But that isn't what anyone suggested.

Instead it is just a small patio. Why can't that small patio be flat?

I see my patio example is causing a bit of debate here. So I take a 4 ft decking post and then dig a hole so that it rises above the ground level 2 ft. I then take a 2nd post, dig a hole and place it in the ground a couple of metres away. I then place a beam on top of the two posts and place a spirit level in the centre of the beam. I then lightly knock each of the posts until the spirit level shows the bridging beam is completely level. With the posts secure and the tops of the posts all level with each other I secure them with concrete.

I then continue this pattern until I have a series of posts secured in the ground, the tops of which are all dead level with each other. Finally I connect all the posts with a framework of beams so that they are all flush with the tops of the beams.

In that case, assuming it is done perfectly, it is level, not flat.