The Surface Area to Volume Ratio or Why Elephants Have Big Ears

There are very few things that are so far reaching across many different disciplines, ranging from biology to engineering, as is the relation of the surface area to the volume of a body. This is not a law, as Newton’s second one, or a theory as Darwin’s evolution theory. But it has consequences in a diverse set of situations. It explains why cells are the size they are, why some animals have a strange morphology, why flour explodes while wheat grains don’t and many other phenomena that we will explore in this article.

What Is SA:V?

All bodies have a volume, and a surface area. It is as simple as dividing the area of the body in question by its volume to obtain the ratio we are interested in. Consider for example a cube of 1 m in side. It has a volume of 1 m³  (1 cubic meter). At the same time each of its 6 faces has an area of 1 m² (1 square meter), and a total surface area of 6 square meters, therefore the surface area to volume ratio (or SA:V for short) is 6/1 = 6 m-1. This ratio varies with the body size, if you do the same calculation for a cube of 2 m to a side, you get a SA:V ratio of 3 m-1, and for a 10 m cube the value goes down to 0.6 m-1. It will tend to zero as the cube gets larger.  The SA:V ratio also depends on the morphology of the body,  for a given volume, the sphere is the object that has the smallest SA:V ratio.

Why the Surface Area to Volume Ratio Is Important

The surface of a body is important in many ways, because many reaction and transfer processes are directly proportional to surface area. Some examples are:

Heat transfer: Heat is transferred to and from a body primarily through its surface, but a large body has less surface available per unit volume (and mass if the body is homogeneous). In the figure below, the larger cube has  54 units of surface and 27 of volume, with a SA:V ratio of 2. The smaller one, has 6 units of surface and one unit of volume, with a SA:V ratio of 6. That means that the smaller cube can be heated or cooled 3 times as fast as the larger one.

Change in the SA:V ratio as a body is scaled. Image from OBEN Science 7E.

Animals need to get rid of excess heat, but this becomes increasingly difficult for big species that have a small surface area compared to their body mass. So, special adaptations are needed in order to cool off a large animal. Such is the case with the elephant in the feature image. The big ears and the skin wrinkles provide the additional surface that is needed for cooling. Small animals have the opposite problem, they lose heat at a very fast rate, so they must eat large quantities of food in order to replace the energy that has been lost as heat.

Steel wool catches fire when heated by the electric current from a battery. Image from Creating Wonder blog

Objects that are very small have a very large SA:V ratio and therefore they heat very quickly. This is the reason why even metals can “burn”, like the steel wool shown in the image or the powder metals used as fuel.

Another dramatic example is grain dust, that can be explosive. The Great Mill disaster is a sad example of that. In 1878, the Washburn A Mill exploded along with several adjacent flour mills, killing 18 workers and destroying the largest industrial building in Minneapolis.

Biology: The cells in every living thing need a set of substances to get into it in order to fuel the cell reactions. At the same time waste products need to be taken out. Cells rely on diffusion through its membrane in order to move substances in and out. As the cells grow, their SA:V ratio becomes smaller, and the membrane area is no longer sufficient to move substances at the necessary rate. Because of this, the SA:V ratio does impose an upper limit on the size of a cell. On a larger scale, several body structures have evolved to maximize the SA:V ratio, such as the lungs and the intestines. Your intestines have no less than 300 m2 of surface area available for digestion.

The SA:V ratio dictates in some way the form and shape of animals. Image from Tall.Life

Engineering: In engineering, the SA:V ratio is also known as the square-cube law. When an object is scaled up by some multiplier, its mass increases as the cube of the multiplier, but its surface and cross sectional area are increased only as the square of the multiplier.

Consider a very large airplane such as the Airbus 380. Its wings are proportionately larger that those of a smaller airplane such as the Boeing 747. Just scaling up the 747 to the size of the 380 will result in wings with not enough surface area to give enough lift for the weight of the airplane. A skycraper that is double the size of another in its dimensions has eight times the weight, but only four times the base area, therefore you may need different construction techniques, such as using steel rather than only wood and brick.

You may have noticed how different very small animals are compared to larger ones. Ants have legs that are very skinny compared to their bodies, and they can lift many times its own weight. Elephants on the other side, have thick legs and they cannot lift great weights compared to their own mass. The reason is the same, when the dimensions grow, the weight increases faster than the area of the legs, and the pressure exerted is significantly larger.

We have given just a few examples of the many instances where the SA:V ratio has an influence. It is hard to think of a discipline of human knowledge where this principle does not apply. In the end, we can say that yes, size does matter.

48 thoughts on “The Surface Area to Volume Ratio or Why Elephants Have Big Ears

  1. I would point out a mistake in the image from OBEN Science 7E. In the 2x cube (center) it says volume = 2^3 = 3. It should be volume = 2^3 = 8. It threw me for a loop when I was trying to get the surface/volume ratio of 3.

  2. This is why smaller mockup models of the twin towers don’t really say much about why it came down (Looking at some of the engineers for justice, the ones not from American agencies putting up fake info in an attempt to make false debunking videos: Looking at you Jessie Ventura!!!!!!).

    However, still, something wasn’t right that day with how those buildings went down.

    P.s. I do believe that nefarious, ill intended terror organizations were behind the crashes etc,
    but as for the building more or less falling into it’s self however, I seem to come to my current conclusion it was a fatally embarrassing* accident, as though they tried to evacuate everyone before they tried to avoid a worse disaster of the buildings falling atop other buildings, but the call was made too early.

    *One of the minor reasons to cover up, that alongside those who won’t want to be hung for such a blunder being a major reason.

    That is as-so-far what I can determine.

    However this Hack A Day article slightly strengthens some of the official (USA Govt funded/influenced) theories by the following, “A skycraper that is double the size of another in its dimensions has eight times the weight, but only four times the base area, therefore you may need different construction techniques, such as using steel rather than only wood and brick.”
    Meaning the down-force from the weight towards the bottom of the building would of been great in comparison to a building half it’s size (such buildings have survived aerospace vehicles of similar ratios plane:Building, likely due to lesser upward mass on such a small base)

    P.s. Being someone whom makes his own opinions from comparing many sides of information and weighs them against known scientifically factual and engineering knowleges, I’m sure I’ll come across some information I can remodel such parts of my opinion, especially since there is a lot of misinformation out there (I can see through most of it, however I am only human and have made mistakes a-plenty)

    1. Not to rehash an old subject. but…. First, I have worked in the steel industry in many areas including quality control – which encompasses metallurgy and grain analysis for many years before I retired. Steel, depending on the type, changes it’s molecular structure at around 700C, and additional structures are created within it at that temperature and above. Structural steel like that of which they would have used in the twin towers is no different. The jet fuel – when combined with other materials like wood, plastic, and certain metals contained in the furniture and walls will burn in excess of that temperature easily, plus you need to look at the airflow cross section at the height of the open area – so you might be talking 50 mph winds or higher – lots of O2, so a forge in other words. So while the steel may look the same to the naked eye, you are re-annealing and changing the molecular bonds and grain contained in the steel – which turns it into a noodle at higher temperatures, no longer being able to hold the structure above it, or itself for that matter. You can argue all you want on the details of the event – but the actual steel in the tower, since it is built to re-enforce itself in the structure (a unibody if you will), once it become non-load bearing anywhere in the cross section, the whole piece comes down. Obviously weight is part of that calculation – as presented in this article, but once again – steel cannot hold it’s own when heated for a period of time – like the twin tower was.

      1. I also already accounted in my own model for what you said, the forge effect is viable in its greatest, the problem of cold spots around adjacent areas would of caused anchor points.
        Those combined with the underneath surface area of cold structural steel. I was expecting at least one of the buildings to topple under the hottest point.
        The downward forces on the steel beams under and above the hot-floor would of suffered a lot of compression at high energy, yet depending on how baked they got.
        This means that the compression effect could effectively melt the steel from the collapsing weight (A transfer of kinetic energy into heat), however the structure directly under would have uneven layers of heat where the entry point and thus the longest burning areas thus would require some strong back-draft mixed with a supply of fresh air to cause such a change in temperature hotspot (post-crash explosion sounds were heard, thus indicating likely-hood of such a back-draft). Cold steel would of acted like catch points and would of caused the top (The cold part of the top) to grip and roll (The grip and roll effect can partially be seen in one of the falls. however doesn’t seem to grip for long and seems to start collapsing in towards the hotspot before seemingly free-falling). The grip and roll would be causable by tangling of the two cold structures and would allow the top half to collapse directly atop other buildings (Due to cross section to height ratio).

        To re-account for the kinetic to heat transfer, that would explain the cold building under the fire and lava-like steel melt becoming warped. Yet again I’d expect the building to warp sideways and crumble the opposite side thus causing the building to essentially snap in half upon collapse

        .
        In other words surrounding buildings were at great risk and thus the buildings collapsing into itself was either a stroke of luck that a large sum less people died that day than could of if the building had collapsed onto other buildings (Causing a domino effect of damage)
        .

        So I would expect one to topple sideways at least half way down due to lower floors having stronger upwards holding force and since lightning don’t strike twice, the other building to collapse in on itself would become very believable.

        Though both buildings collapsing in a neat pile…. Could be an act of god so to speak, lesser people died than what could of, even though ideally the incident should of failed to happen and no-one should of died.

        (p.s. The above is only a small 10-ish % of what I have thought through about the incident)

        1. Ok, a lot less than 10% of my current knowledge after recalling some more things I have read/seen/experimented/learnt in college(BTEC L1&L2)/etc, Depending on what I currently have on my top of the thinking priority list (A mental list of importance determining, per situation dependent).

        2. I thought about the pile thing a lot. I believe that the kinetic energy involved and just plain weight would negate the “sideways” effect of the falling structure. I believe at those levels things flatten more than fall – things like steel would actually heat from the crushing effect – how much I don’t know, but I can guess quite a bit. Just as it would pull in lower temperatures as it decompressed. Once again – there are no comparative things that we can use to extrapolate any more that a fraction of the forces going on in a collapse like this. I suppose it’s possible that the structure hade a much higher heat level in it’s center and fell more inwards, with the outer shell holding a little bit longer in time – like water going down a drain in the center, maybe even with a twisting effect internally.

          1. The pile effect is also a likely cause as well, some ifs and maybes, however this seems the likeliest.
            The usual videos collectively did show some twisting effect as the building fell, thus showing yet again the likeliness of this effect. However the “core” (The heavily structured part) is actually several seperate smaller cores around the outside, like pillars (So I have heard from both official stories and the EFJ groups), they are supported by a central pillar of thicker than usual steel*. This would lead me to expect the building to try and pivot under said center. Though the surrounding pillars may have been densely packed enough of structure to even out the fall and cause the pile effect (Yet again guess work)

            *I haven’t seen the official blueprints yet so I cannot confirm these claims, even to myself yet

            .
            .
            For a note, Have you seen that Seconds From Disaster show where they show a simulation of a plane whom it’s alloy wings cut through the tensile steel beams with minimal breakage of said wings?
            Like they were trying to claim you can, “cut a knife with butter” (So to speak). Makes you wonder who talked more rubbish about the towers: The official stories or the Tinfoil Hatters? (They both are as guilty as each other over this incident)

        3. I have not seen “Seconds from Disaster”. The Wings cutting through steel is a funny one. I think they believe that the tornado effect of high speed objects counts when it comes to structural steel. I suppose it would if you could get the wing up near 20k mph – maybe a few scratches.

        1. It’s why I entertained an answer to this subject in the first place. The surface structure analogy of the twin towers is an interesting thing to me. I’m betting that the temperatures were much higher than the 700C we normally associate with jet fuel burning – towards the center of the “forge”. I’m not sure cold points count here, part of the problem is I’m not a structural engineer. I know they don’t build things symmetrically – like bridges because of harmonics, and I’m guessing that a skyscraper that needs to move 20 + feet in a given direction in the wind is designed to move at those points. While I can explain the changes in the steel and the formation of other structure within it at higher temperatures – I can’t tell you the weight bearing of the steel they used – nor the torsional stress that would accompany heating a structure with colder areas within it. It’s all pretty fascinating stuff – but hard to prove without enough logging equipment embedded in the building itself.

      2. Knowing about this behavior of steel already, my first thought, when I saw the pictures of the burning buildings in TV was:”That will come down. At least the one with the fire lower than the other. (more mass/weight above)”
        Once I had an experience with a piece of steel rod (around 12mm thick) which somebody used to stir a campfire and left it there afterwards. When I took it, about 20cm of it were glowing very dark red, not brightly visible even in darkness. When I used it again it bent over like solder – no strength left.
        So I don’t need any conspiracy theories, it’s just physics.
        Another question would be, who could have known beforehand or not and what information secret agencies could have gathered in advance from their mass of data or not.

        1. I came across some unofficial hacking stories. Apparently it was two kids, one a son of an employee of the plane manufacturer whom both kids (Teenagers) were being groomed to hack for Al Qaeda by pretending that the hacking is “for competition” and started to up the anti by payments etc…. Even they seemed to suspect the groomer was a fake undercover CIA/FBI agent….. I can’t remember where I got said PDF, however by the time I find it, ya’all lose interest of this or forget this article’s comments by then…

          However I came across snippets of information here and there about Al Qaeda using and tricking young hackers to give al qaeda inside information. This gives me a feeling that Al Qaeda were behind the crashing, however this alongside other info points to the American (USA) 3 letter agencies being aware of the pending attack…. Seems they acted too late for the crashes, could of been prevented, hence the increased security and the lesser freedom as a result (Though it didn’t affect freedom by much when compared to the type of security measures at the time, now however…. Privacy issues and freedom problems galore! even outside USA long after since then).

          Though alternatively, I can see why some people paint said 3-letter agencies with a guilty face, Said blunders can easily be misinterpreted as “an inside job”.

          Though “an inside job” phrase can still be applied (If true) to the tricked hackers, whom were “insiders” in the sense they are living in the USA when they gathered the information (Still suspicion around if it was an anti-hacker sting or not).

          Blah, Blah, So on and So forth (Could be here forever about this side of said story)

          1. Forgot to mention, as you mentioned, the lower crash: I would’ve more expected that building to collapse in on itself. That is due to higher mass above the fire.
            However the other building, I’d expect the top being lighter (relative to the imploded building) to fall sidewards.

            Also the lower floor fire building one… Remind me please, did the building with the fire on the lower floors fall first?
            (As I’d expect as the pressure above the fire was higher and thus more stresses, I’m gonna refresh my memory soon, even if no reply)

  3. One of my electronics instructors told us in class, “If they ask you during a job interview; “What is a heat sink?”, don’t tell them, “That is where you dump the heat.”,”

    B^)

    1. Now we need a way to turn off hunger to get skinny and cut A/C use and tolerate heat. This “square-cube law” also explains why that little bit of coffee gets cold instantly, resulting in wasted coffee as well as coffee being too cheap. If coffee cost $100/gallon, people would finish every cup.

    2. My observation is that as the weather cools down, the fat people are usually the ones to notice first. Most of them are fat due to a low metabolic rate. There’s way more variance with skinny people. In fact, I have noticed that among my friends, the ones who are least sensitive to cold weather tend to be on the skinny side and eat quite a bit more than you would expect. They also tend to glow brighter on a thermal image.

    1. … and you don’t want to be sitting at a radio checkpoint when a horse walks up and decides to dump the contents of his bladder right in front of you. (With highly embarrassed young female rider on his back saying “I can’t make him stop!”)

  4. Suggest a video with the steel wool and 9v battery instead of just that pic, else yank the pic. Folks may think it’s the itty bitty flames in the pic you’re talking about but then find the whole huge roll furiously burning then eating it’s way through the carpet. Observed a demonstration of this in USAF, mandatory required training session before allowed to move the family into base housing. Most impressive!

  5. Many architects just don’t get this important concept so when they say we should build smaller houses in terms of floor area and you say no you need to design houses with a smaller surface area they can get very cranky and call you a global warming denier or whatever. Never work with innumerate fools.

    1. Lot of people don’t get this concept, it’s one of those very basic science facts that a lot of people don’t get, even when you take them through the demo with blocks like shown above. I’d mention some other ones, but it’s almost like deliberately trolling, because it will get a dogpile of “No! Common sense says that…” and “Stands to reason that…” … ugh…. I’ve even had “I’m an engineer and….” WTF??? Remind me to stay the hell away from anything you designed.

  6. “You may have noticed how different very small animals are compared to larger ones. Ants have legs that are very skinny compared to their bodies, and they can lift many times its own weight. Elephants on the other side, have thick legs and they cannot lift great weights compared to their own mass. The reason is the same, when the dimensions grow, the weight increases faster than the area of the legs, and the pressure exerted is significantly larger.”

    Somewhere in there’s the reason for walking on fours vs twos.

    1. Yeah but if you’re working with a basic 4-limbed genome, and you need wings or arms, then you have to compromise. It limits how big a biped can be, but it didn’t do tyrannosauruses any harm.

      Weird actually how insects and other creepy things manage to have plenty of legs, but vertebrates are stuck with the basic 4 limbs. I suppose there’s a lot more genetic variance in the creepy crawly world.

  7. This is also the reason why we cant have giants. Gallileo proved that the mass of the giant would rise as a cube of its size while the flexural strength of its bones would only rise a square (proportional to the cross section of the bone). Poor giant would get crashed under its own mass :(

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