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A counterpart to the question Why doesn't the Falcon-9 first stage use three legs to land?

This excellent answer states that the first stage of the Falcon-9 uses four legs because four legs are less likely to tip over. However Starhopper which is a prototype for Starship has three legs, as seen in this 150 m Starhopper test. Furthermore, four legs mean that its apothem is larger meaning it will be more stable than three legs. In order for three legs to have the same stability has 4 legs, the legs have to 1.4 x the distance from the center, so Starship might be lighter and cheaper with four legs as it can have smaller and lighter landing legs.

The answer to the question Why did the Apollo Lunar Module have four landing legs? states that

Three legs offers less safety margin for steep slopes and sideways landings.

Unlike the first stage of the Falcon-9, Starship is not guaranteed to land on a perfectly flat surface on Mars. It's entirely possible that Starship can land on an uneven surface, thus needing more stability and the need for four landing legs.

Question: So why does Starship have three legs when it's going to land on Mars? Is it simply weight, or another reason?

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    $\begingroup$ Starship can hover. $\endgroup$ – Bob Jacobsen Aug 28 at 16:55
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    $\begingroup$ What's the point you're trying to make? $\endgroup$ – StarMan Aug 28 at 16:56
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    $\begingroup$ The "1.4x" holds for a thin and long body, but not for a short stump like this. I guess it couldn't tip over even without any extending legs. And: This is "mockup" never going to land on Mars. $\endgroup$ – asdfex Aug 28 at 16:58
  • $\begingroup$ F9 has to land in a predefined spot; it’s trajectory ends in a pre-chosen point. Like Apollo 11, Starship can hover and look for a better landing point. But I have no idea whether that was a design criterion. “Why” questions are hard to answer unless one was there. $\endgroup$ – Bob Jacobsen Aug 28 at 16:58
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    $\begingroup$ @asdfex Nope, it holds true for a circle (or cylinder), regardless of height. And Starship is going to have (most likely) 3 legs. $\endgroup$ – StarMan Aug 28 at 17:03
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In the question you mention about why Falcon 9 uses four legs to land instead of three, there are multiple great answers.

One of them by @David Hammen states that:

"Another factor is that the Falcon has nine engines, one in the center and the other eight around it in an octagonal or square-like arrangement. The bottom part of the thruster reflects this tetradic / octagonal symmetry. The bottom attachment points are close to the engines and have to fit within the constraints of the engine layout. Having four landing legs obeys this symmetry; having only three would not."

Latest Starship architecture has an hexagonal pattern of seven engines, a central one, and six at the hexagon's vertices.

The reasons for the choice of the number of engines prevails over the reasons for the choice of the number of legs.

The choice of the number of engines dictates the shape of the engine's mount, and legs are easily attached to engine mount's shape.

Therefore in this case, it is strucurally more convenient, thus lighter, to go for three legs attached to an hexagonal engine mount.

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    $\begingroup$ So, it's basically 3 legs because dividing 6 sides by 4 legs is more problematic than dividing 6 sides by 3 legs? That makes perfect sense. Sometimes, even rocket science isn't rocket science. $\endgroup$ – Mast Aug 30 at 10:01
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By the time Starship launches for Mars, SpaceX may have changed the number of legs several times. They redesign as quickly as the Haggunenons of Vicissitus Three evolve during lunchtime. Plausible guesses for factors favoring three legs, at the moment:

  • lighter
  • less aerodynamic drag, if the legs also act as fins
  • send a scout beforehand to choose a boulder-free landing zone
  • a human pilot on board can finesse the landing zone better than a NASA robot's "seven minutes of terror"
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    $\begingroup$ +1 for the Hitch-hiker's Guide reference $\endgroup$ – Organic Marble Aug 28 at 20:43
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    $\begingroup$ Another possibility for your list - three was the minimum to be stable at all, and a fourth would have required more time spent on welding before they could fly. $\endgroup$ – Phil Miller Aug 30 at 20:32
  • $\begingroup$ One is the minimum to be stable, if you have Kerbal Space Program's oversized reaction wheels and you've landed on a low-gravity body. I'd attach a pic of my bungled landing that tore off a few legs, but you get the idea. $\endgroup$ – Camille Goudeseune Aug 30 at 21:30
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An answer might be, that three legs will always be able to land on a rough surface with all three legs touching the ground, while with four legs there is the possibility that only three of them will touch the ground. In order for the four legs version to touch the ground with all four legs, those legs should be made telescopic with some sort of dampers, making the legs heavier.

The Starhopper depicted in that video doesn't seem to have telescopic legs from what I see, but I didn't dig deep into this.

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    $\begingroup$ ... and with four legs, without telescopic adjustment most of the weight will be carried on a diagonal pair, meaning that each leg would have to be 50% stronger than a leg on a tripod $\endgroup$ – JCRM Aug 29 at 9:12
  • $\begingroup$ +1 - Starhopper was landing on the ground, with no swinging on the weaves but with potential unevenness. So less reason for trip prevention, more reasons to prevent wobbling. $\endgroup$ – Mołot Aug 29 at 9:22
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As @camille-goudeseune writes, we shouldn't draw any conclusions from this "flying water tower" towards the design of the final Starship which will be an order of magnitude larger. Nevertheless, we can have a look why it makes sense to have three legs on Starhopper:

It's true that the three legs need to extend 1.4x from the center of the body of the spacecraft compared to four legs. If we assume the body to be 1 unit radius and the legs extend to a distance of 2 units from the center (seems somewhat reasonable given the linked footage), we can calculate that the radius of four legs would be 2/1.4 = 1.4 units. I.e. they need to extend by 0.4 units from the edge of the body compared to 1 unit in the three leg case. That means they could be 60% shorter, not a mere 30% as in the Falcon case.

However, it does not mean that the legs can be shorter and lighter by this amount. The legs of Starhopper seem to consists of two struts and some thin filling material in between. The two struts are mounted at the upper and lower rim of the body. It's likely that this is the only place they can connect to and that the walls of the tank are not stable enough to support the landing leg.

So, the two mounting points of the legs can't move, only the foot can get closer to the body if we use 4 legs. This means, that the length of the struts (that are already quite steep) won't change much, calculating the geometry of the triangle gives a reduction of less than 10% - not enough to make up for the additional weight of a fourth leg.

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You question assumes the stability of the craft should be the driver for the positioning of the feet, and hence the legs.

If instead, aerodynamics is driving the design of the aerodynamic surfaces, and they are being reused as legs, then provided the stability they afford is more than sufficient then there is no need to optimise for stability.

(NB, This answer does not mean to imply the existence of intelligent design)

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    $\begingroup$ I think a lot of people have forgotten that two of Starship's three "legs" are actually movable control surfaces that will simply be reused at landing (at least, according to last year's design; we'll see what Musk says in a few weeks). You could probably get away with four flaps/wings/legs, but their current position and design is driven first by a need for supersonic aerodynamic control surfaces, and only second by the need for something to keep Starship's butt off the ground. $\endgroup$ – CBHacking Aug 29 at 1:15
  • $\begingroup$ @CBHacking Due to its larger size, Starship is designed to re-enter belly first for more surface area (unlike Falcon 9 which re-enters butt first.)You can see when Starship comes belly first through the atmosphere, that the side fins are folded back and the back fin also maintains the belly first orientation. The last thing you want is a fourth, belly fin, which would leave Starship with no preferred orientation. I am amazed that this answer does not have more upvotes, as it is clearly the right answer. $\endgroup$ – Level River St Aug 29 at 22:47
  • $\begingroup$ space.stackexchange.com/a/9199/6117 shows the distance from the edge of the square formed by Falcon 9's feet to the centreline is about 1/6 of its height. Distance from the edge of the triangle formed by Starship's feet to the centreline seems a little less than 1/6 height (but not 1/1.4 times less) so by this measure Starship seems perhaps slightly less stable. But it is a larger vehicle so will be less affected by the wind. BTW, I'm sure the number of legs on the booster isn't finalized. It will be taller (but will not be required to re-enter the atmosphere at full orbital velocity) $\endgroup$ – Level River St Aug 29 at 23:01
  • $\begingroup$ but will it be required to land on a barge @LevelRiverSt $\endgroup$ – JCRM Aug 30 at 7:12
  • $\begingroup$ @LevelRiverSt: I think it is actually the other way around. Only two of the "legs" are aerodynamic, the third one could actually be "just a leg" but only looks like a fin because of symmetry. Unlike the other two, it is however not movable. And it is the two side fins that are aerodynamic, and the top fin that doesn't do anything: youtu.be/zu7WJD8vpAQ?t=2370 $\endgroup$ – Jörg W Mittag Aug 31 at 19:41

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