How can ants carry items much heavier than themselves?

Strength is proportional to a surface area divided by volume, but since volume is directly proportional with mass and I can't get an accurate density (I am guessing approximately both for mass and size.), I will use mass instead.

According to Wolfram Alpha, the average mass of the human body is 70 kilograms. The surface area of a person weighing 70 kg with a height of 170 cm is 1.818 square meters. This gives us a weight/surface area ratio of about $38.5 \frac{kg}{m^2}$.

So now, how much does an ant weigh?

This article provides a variety of different numbers, varying from 1 mg to 60 mg. Since the biggest ants will be soldiers, I assume that the approximation will be slightly smaller than 30 mg. Say 25 mg or 0.000025 kilograms.

Now comes the interesting part. Not Wolfram, not even uncle Google knows the surface area of an ant.

This Britannica page says that ants range from 2 mm to 25 mm. Let's eliminate the soldiers since they are huge. (A big worker would be as long as 8 mm.) That gives an approximation of 5 mm.

I gave the animation industry a shot and tried to measure the surface area of this free ant model. The length of the ant is now 0.005 - let's call it a meter.

This gives us a surface area of about $4.87\cdot 10^{-5}$, or $0.0000487$ square meters. So an ant that weighs 0.000025 kg with a length of 5 millimetres has a surface area of about $0.0000487 m^2$. This gives a weight/surface area rate of about $0.5335 \frac{kg}{m^2}$.

So, the uniform strength of an ant is about thirteen times more than a human's.

How much can a human carry while walking a long distance, maybe even climbing? Maximum 20 kilograms for most people. That is slightly more than a quarter of our weight (about 0.28).

How much can an ant carry? About 1 gram - the weight of a leaf, or 40x the weight of an average ant.

4 divided by .28 = 14. So ants are about 14 times stronger than we are. (Carrying capacity according to body mass.)

I think the answer has less to do with their construction and more to do with their smaller size

For more information lookup Scaling Laws.

Basicly the mass of a object scales as it's size cubed so a ant 10 times the size will be 1000 times heavier. But the strength of an organism depends on the cross sectional area of muscle (I've heard this somewhere, not sure about the details), and hence scales as the size squared. So an ant 10 times the size will only be 100 times stronger.

Putting those two facts together the strength to weight ratio of an organism varies inversely with it's size. Hence smaller organisms even with the same construction will be able to lift more in relation to its mass.

Note: When I say size I'm referring to the linear size of a body as measured with eg a ruler

Strength / weight is a funny thing. The stress on a long thin rod (like an ant's leg) is limited by the Buckling strength which is given (for rod that can freely rotate at each end) by

$$F = \frac{\pi^2EI}{L^2}$$

where $I$ is the second moment of area which scales with $r^4$ - so

$$F \propto \frac{r^4}{L^2}$$

So when you make an object 2x smaller, the mass is 8x smaller but the strength is only 4x smaller. This means that smaller objects are stronger for their weight.

AFTERTHOUGHTS

Ants have an exoskeleton meaning that their legs derive most of their strength from the outermost part of their body (think "skin as tough as bone"). This makes the "second moment of area" of the support structure much larger than you would expect - see that $r^4$ term above... This is one reason why the skinny legs of the ant are quite so strong - all their strength is on the outside.

Having established that the (exo)skeleton of the ant has greater structural strength, weight for weight, than that of larger species, we still need to address the question of muscle strength. Here we need to look at the surface-to-volume ratio. Doing work with a muscle requires oxygen - which is obtained by exchange of oxygen with the atmosphere. Now if we assume that the volume of muscle scales with the volume of the animal, and thus with $r^3$, and the surface area of the lungs, or spiracles in the case of ants (tubes from the skin to the muscles) scales as $r^2$, then you can see that the "lung to muscle ratio" (LMR) is

$$LMR \propto \frac{1}{r}$$

so the smaller you are, the less likely you are to run out of breath. Even if the lung is a fractal surface with a fractional dimensionality greater than 2, it will be less than 3 and the LMR is still larger for smaller animals. Diffusion of oxygen - same story, because it has much less far to go.

In short- by dint of their size, the structure of an ant is more resistant to buckling; and their metabolism (ability to burn oxygen) is better which means their muscles can work harder.

Clever little things, really.