I know the earth is a sphere, I’m definitely not debating that, but I also know there are various tests you can do on a relatively small scale to prove that the earth is not flat,. Are there similar tests that prove specifically that it is a sphere?
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Not exactly backyard-backyard, but close enough.
First, some background on a torus. In essence, it’s a cylinder where you tape the top and the bottom together. As such, it has 2 relevant radii: the (typically) small radius of the cylinder (let’s call this r), and the height of the cylinder which becomes a (typically) larger radius in the centre of our torus. See also, [this picture](https://upload.wikimedia.org/wikipedia/commons/8/81/Torus_cycles.svg). For this discussion, I’ll use two other radii instead of the larger one: the radius from the centre of the hole to the closest surface of the torus (R’), and the radius from the centre to the furthest surface (R). Or, [visually](https://upload.wikimedia.org/wikipedia/commons/1/17/Tesseract_torus.png), the radius of one of the green or red circle is r, the radius of the smaller blue circle is R’, and the radius of the larger red circle is R.
If we look out towards the horizon, then we can only see so far, because the Earth curves. Even if it is a torus. And we can figure out how far the horizon is based on how high a vantage point we have, and how big the radius of our planet is. In a first approximation where we don’t go too high, [the Pythagorean theorem](https://upload.wikimedia.org/wikipedia/commons/2/21/GeometricDistanceToHorizon.png) is all we need:
(R+h)²=R²+d²
with R the radius of our planet, h how high up we are, and d the distance we can see. Rearranging this for d² gives us:
d²=h*(2*R+h)
If h is very small compared to R (ie meters compared to kilometres), this can be approximated as
d=(2*R*h)^(0.5)
The important part of this equation is that the distance the horizon is at depends on the square of the radius of the planet.
How do we use this to figure out if the Earth is a torus? First, assume it is. What would happen if you looked out at the horizon? There are 3 extreme cases I can think of:
1. you’re standing on the circle with the largest radius (ie the large blue circle in the second pic I linked)
2. you’re standing “on top” of the torus (ie the orange circle in the same pic)
3. you’re standing on the circle with the smallest diameter (ie the smaller blue circle)
Of these, the first one is rather boring, so we’ll start with that. In that case, there are two other extreme situations: you’re looking directly along the larger circle, or directly along the smaller one. Either way, the horizon will be at different distances. We can use this by looking measuring the horizon in two different directions that are less than 90° apart (preferably 45°). If both measurements are significantly different, then we live on a torus (for Earth, the difference between the equatorial and polar radius horizon is about 8.5 metre for a 2m tall person)
For the second one, well, in one direction, there simply is no curvature. The orange circle lies flat on the torus, so there is no curvature in that direction. Meaning you would be able to see everything there (disregarding atmospheric effects). And you should even be able to see where the torus curves to the left or right.
And similarly, for the final option, you should be able to see the planet curving upwards. it also has no horizon, and in a best case scenario, you should be able to keep following the curve upwards to its zenith, and then down again!
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