# Why are four-legged chairs so common?

Four-legged chairs are by far the most common form of chair. However, only three legs are necessary to maintain stability whilst sitting on the chair. If the chair were to tilt, then with both a four-legged and three-legged chair, there is only one direction in which the chair can tilt whilst retaining two legs on the ground. So why not go for the simpler, cheaper, three-legged chair? Or how about a more robust, five-legged chair? What is so special about the four-legged case?

One suggestion is that the load supported by each leg is lower in a four-legged chair, and so the legs themselves can be weaker and cheaper. But then why not 5 or 6 legs? Another suggestion is that the force to cause a tilt is more likely to be directly forwards or sideways with respect to the person's body, which would retain two legs on the floor with a four-legged chair, but not a three-legged chair. A third suggestion is that four-legged chairs just look the best asthetically, due to the symmetry. Finally, perhaps it is just simpler to manufacture a four-legged chair, again due to this symmetry.

Or is it just a custom that started years ago and never changed?

• There are 1-legged chairs. – Mike Dunlavey Jan 24 '16 at 14:32
• "However, only three legs are necessary to maintain stability whilst sitting on the chair." This statement is simply false, a 3 legged chair is considerably less stable than 4 legged chair. – JamesRyan Jan 25 '16 at 12:26
• Yes, a 3-legged chair is less stable than a 4-legged chair. But the statement is that a 3-legged chair is stable whilst sitting on the chair. Here, stability is used as a binary term, rather than a continuous measurement. In this way, if you sit on a 3-legged chair in a typical manner, the chair will be stable. – Karnivaurus Jan 25 '16 at 12:54
• Moreover, 4-legs have the problem that when one leg is a bit shorter/larger, it will be unstable – becko Jan 25 '16 at 13:43
• @JamesRyan don't you know that 3 points make a plane? a 3-legged chair will never toggle around like 4-legged chair – phuclv Jan 26 '16 at 16:08

Suppose the leg spacing for a square and triangular chair is the same then the positions of the legs look like:

If we call the leg spacing $2d$ then for the square chair the distance from the centre to the edge is $d$ while for the triangular chair it's $d\tan 30^\circ$ or about $0.58d$. That means on the triangular chair you can only lean half as far before you fall over, so it is much less stable. To get the same stability as the square chair you'd need to increase the leg spacing to $2/\tan 30^\circ d$ or about $3.5d$ which would make the chair too big.

A pentagonal chair would be even more stable, and a hexagonal chair more stable still, and so on. However increasing the number of legs gives diminishing increases in stability and costs more. Four-legged chairs have emerged (from several millennia of people falling off chairs) as a good compromise.

• Here's the Vehicle situation ! youtube.com/watch?v=QQh56geU0X8 – Fattie Jan 24 '16 at 16:19
• 4-legs has an advantage over 5+ legs on irregular floors as you can always rotate the chair to a point where all four legs are on the ground. You can't do this with 5+! – ejrb Jan 25 '16 at 15:03
• @ejrb That has the unstated assumption that the ground is uneven to at least the same degree as the difference in the table legs' length. For example, if you have a table with legs of length 1m, 1m, 1m and 0.5m but no two points on the ground differ in altitude by more than 0.25m, the table can't be placed securely. – David Richerby Jan 26 '16 at 19:29
• @DavidRicherby Indeed, my comment was just meant for flavour as it is outside the constraints of the question, and does come with the caveat that the legs need to be equal lengths. If you've got wonky legs your best option is a tripod (unless your floor unevenness exactly counteracts your leg wonkiness of course!) – ejrb Jan 27 '16 at 16:11
• @ejrb It's quite a while since i watched that excellent video so I'd forgotten it assumes equal-length legs. But, now I think about it, that's an obvious requirement for the proof that's given. – David Richerby Jan 27 '16 at 16:36

The real question is why not 2 legged chairs?

First, our legs are single points that contact the floor. How they reach the floor is not important.

If you have $n$ legs evenly spaced, then the ratio between the radius of the legs and the length of the shortest axis is $\cos(\pi/n)$:

• 2 legs: 0
• 3 legs: 0.5
• 4 legs: ~0.71
• 5 legs: ~0.81
• 6 legs: ~0.87
• 7 legs: ~0.90
• 8 legs: ~0.92

So for a given chair radius/diameter, your ability to tip the chair is measured by the above value. A lower value means it is easier to tip the chair along its "tippy directions".

• 0 legged chairs are known as sitting on the floor.
• 1 legged chairs don't even have a non-tippy direction. They are apparently used when milking cows.
• For a 2 legged chair, it is infinitely tippy along its tippy direction. So clearly that is a bad plan. You can balance by using your legs as a 3rd/4th leg.
• A 3 legged chair has half the stability along its tippy directions as it does along its long direction -- towards a "face" as opposed to a "corner". You can "splay" the legs to make up for it, or accept low stability.
• A 4 legged chair has 71% of the stability towards its tippy direction.
• A 5 legged chair has 81% tippy stability. As an example, rolling office chairs.

The return on additional leg for a 3 legged chair is infinite. For a 4 legged chair, the chair is 41% more stable along the tippy direction than a 3 legged chair. For a 5 legged chair, the chair is ~14% more stable along its tippy direction. (For a fixed radius)

Adding that 4th leg adds a whole bunch of stability. Adding a 5th leg only adds a small amount (1/3 as much). In fact, in order to get the same stability increase from 3 to 4 after you have 4 legs, you need an infinite number of legs (a solid circle or ring of legs).

So 4 legs is the sweet spot between 0 and O.

• We should use chairs with one, but extremely-large leg (as an approximation of infinte legs on a circular perimeter). Consider the infinte return! – loa_in_ Jan 24 '16 at 23:18
• We do. Though I'd imagine some people still manage to tip this particular type of chair. ;p – elmato Jan 26 '16 at 3:09
• @loa_in_ kick stools are usually circular with a larger base than the sitting area. – Viktor Mellgren Nov 10 '17 at 9:32

Five legged chairs are common too.

Sometime in the 1980s a lot of office chairs were replaced - on the grounds that four-legged swivel chairs became quite unstable when the tilting/reclining feature shifted the centre of gravity.

One reference I've found is this, which mentions an Australian safety standard from 1990 I'm pretty sure the mass replacement I described was due to a similar incoming standard in the UK but can't find links at the moment.

Since then, virtually all swivel chairs that I've seen have had five legs.

• I think part of the issue was if one leg broke, on a 5 leg chair it tended to remain upright for long enough to let someone get off. Also if someone leans back on a 4 legged office chair, it is possible to put all the force on a single let. – Ian Ringrose Jan 25 '16 at 10:36
• Mostly it's an issue having to do with the swivel rollers, which both make the effective "diameter" of the leg polygon smaller and add a considerable degree of unpredictability. I can recall some fairly close calls on four-legged rolling swivel chairs. – Hot Licks Jan 27 '16 at 2:40
• @Hot Licks There was a comment to that effect a week ago, but it disappeared. – Brian Drummond Jan 27 '16 at 11:02
• @BrianDrummond - Yeah, at least one of those comments was mine. I suspect that someone doesn't like me. – Hot Licks Jan 27 '16 at 13:55

I'll bet you've never sat in a three-legged chair. They are not very stable. It is easy to tumble out if you lean over too much. I think the problem is that the axis of rotation for tumbling (line between two legs) is closer to the center in a three-legged chair (compared to a four-legged chair) unless the chair is very wide. Leaning over can put your center of mass on the wrong side of that axis causing a torque in the direction that would knock the chair over.

John Rennie provides an excellent explanation as to why a higher amount of legs provides greater stability and improved balance. As for the question of why not greater than 4, it can be answered by considering the environment the chair will be in. It's rational to assume that not all surfaces are equal and flat. As geometry proves, a chair can make contact to the floor with all 4 of its legs on any surface, no matter how warped or saddled, by rotating the chair less than 90 degrees. If you think about this, it makes sense. Numberphile (the youtube channel) has a nice explanation in a video entitled, "Fixing a wobbly table." This doesn't work with any more than 4 legs. Thanks Euclid!

• Nah, it's just that above four there is only marginal improvement, because of, the general ratios of the human body. For certain what we would consider extreme (alien!) body forms, in fact five would be better in a worthwhile way. – Fattie Jan 22 '16 at 3:36

This should be more of an engineering/economics question than a physics one. The statement "... the simpler, cheaper, three-legged chair?" Is false. 120 degree angles have historically much harder to manufacture than 90 degree angles, the small cost of an extra leg is easily offset by the increase in manufacturing costs have having to cut innumerable 120 degree angles.

• interesting point of view ... The same reason holds for 5 etc – user46925 Jan 25 '16 at 16:16

Stability is a valid point, but probably not really the real reason.

People think and design in 90 degree angles. Everything around us is squarish. Rooms, houses, windows, doors, even books and screens. Orthogonality makes calculations and manufacturing simple. It's all about parallel and perpendicular lines, easy cutting, going along the wood grain, zero waste after cutting, and due to all the history and objects around us, the first (and usually the only) design we really think of. Our raw building blocks are usually long rectangular objects: when you take two planks of wood and nail them together, you get a right angle (unless you really took your time to cut and align things in another way).

Honestly, how many people would even consider making something other shape than rectangular or circular? A hexagon is already a stretch, and more trouble than it's worth -- you can't cut in a honeycomb grid because a finite length of the blade prevents you from making a 120 degree turn without ruining the outside. So you're stuck with cutting a rectangle and trimming it to a hexagon.

All in all... it's mostly just geometric simplicity, which makes the engineering much easier for right angles.

• This is true, however, most "rectangular" chairs are not perfectly rectangular, if you look closely. They typically have curved edges to the seat, or some curvature through the seat's surface. Traditionally, it makes sense that rectangular chairs were easier to manufacture, but modern rectangular chairs tend to be much more elegant in design, and so there is wasted after cutting. – Karnivaurus Jan 27 '16 at 11:34

Because the seat is usually square.

If it was round (or another shape) it would be harder to make and harder to attach the back (or extend the back legs to make a back).

If it had no back, it would be a stool. Stools can have three legs.

(Not all answers have to do with physics!)

As explained by John Rennie, 4-legged chairs are more stable than 3-legged ones. It only remains to point out the advantage of 4-legged chairs over $n$-legged ones, $n>4$.

It turns out that on uneven surface only 3-legged objects never wobble, and only 4-legged ones always can be rotated so that they won't wobble. Here's the the abstract of the above paper:

We prove that a perfect square table with four legs, placed on continuous irregular ground with a local slope of at most 14.4 degrees and later 35 degrees, can be put into equilibrium on the ground by a “rotation” of less than 90 degrees. We also discuss the case of non-square table s and make the conjecture that equilibrium can be found if the four feet lie on a circle.

BTW, I find it quite amazing that a result of so basic and of such practical applicability wasn't found until recently.

Perhaps because the surfaces chairs (and tables, beds, etc) are placed on are flat by design with respect to earth's gravitational field. This makes for much better living/working/playing/etc quarters as every current and historical example proves. Building rooms/floors/houses/etc with flat surfaces automatically accommodates any four legged chair/table/bed/etc. Also people/children/balls/etc are balanced on this flat surface. Square chairs and rectangular tables are also much simpler to arrange. So a one-off effort for multiple long-term gains that has existed since before anyone even thought of making a chair. And if you want to sit down while you're fishing from a slanted river bank... there is a solution.

• Only a 3 legged chair can be placed exactly on a flat surface. A 4 legged chair can only ever sit perfectly on an curved surface - and even then only in a single orientation – Martin Beckett Jan 20 '16 at 23:43
• @MartinBeckett Hmm? An n-legged chair can trivially sit on a flat surface, if the legs are the same length. It is true that if they differ more than they flx, I suppose? – Yakk Jan 21 '16 at 0:20
• @Yakk only in a platonic ideal. Any microscopic difference in length or flatness means it will rock. That's why tripods have 3 legs – Martin Beckett Jan 21 '16 at 4:33
• @MartinBeckett you are assuming it's perfectly rigid. In reality items will have some flexibility which will make up for "microscopic imperfections in length and flatness". – Peter Green Jan 21 '16 at 12:45
• All this theory ... a real floor is not flat, and a real chair does not have four legs the same length. – garyp Jan 21 '16 at 16:22

A three legged chair can never wobble, even on an uneven floor, because any set of three points is always coplanar. However... chairs are mostly used on flat floors, so this is not as much of an advantage as one might think over the four-legged chair.

Above all, our bottoms are not triangular. The seat part of the chair is rectangular to match our bum and the legs and feet are generally positioned at the extremities of the rectangle; namely the corners. This ensures that that provided our centre of mass is above the seat, and we are on a level surface, it will act downwards between the feet and the chair will be stable.

However, with every leg we add greater than three, we increase the probability that the coplanar feet will deviate significantly from coplanar points on an uneven floor. In fact the probability of a noticeable wobble doubles as we move from the four to the five legged chair. Furthermore, the fifth leg and any subsequent ones, add nothing to stability since the extremities of the seat are already supported.

The exception to this rule would be a chair on which we may want to lean our bodyweight outside of the four legs, or on which the position of the legs may be unknown. In fact in such instances, such as on an office swivel chair, the chair will often have five castors. These more closely approximate a circle, thereby permitting us to push laterally in order to roll in any direction with stability, whereas a four-castor swivel chair would deliver unpredictable stability depending on whether you happened to push laterally towards a leg or towards the space in-between legs.

## protected by Qmechanic♦Jan 21 '16 at 10:53

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