The hero saves the captured giant, and as a reward, the giant presents the hero with a magic ring... However, the ring is made for giant fingers, so the hero decides to wear it as an armband instead.

If the ring is a suitable size to be used as an armband, and assuming the giant has human proportions, just larger, how large must the giant be?

Clarification:

I am struggling to work out how big the giant must be for this scenario to work. It seems that a finger is maybe 1/10th of the radius of a forearm, but does that mean that the giant is 10 times the size of a human? Does it work that way, or does the radius of an appendage scale up at a different rate to the overall size of a creature? I'm a bit stuck.

EDIT: 1:10 appears to have been a significant overestimation on my part. 1:3.6 to 1:4 seem to be the more probable ratios, as suggested by Trish and Chasly.

Square-Cubic Law tells us that at some size the neck of our giant breaks under its own weight⁽²⁾, but how large is a giant that has a ring that fits a forearm?

Our Giant's finger's diameter is the diameter of the forearm of the human. I am a rather slender person of 2 meters height, my wrist is pretty much skin over bone. To fit my wrist, that'd be $d_w=70 text mm$, while my ring finger is $d_rf=22 text mm$. To slip the ring over the hand (which means I have to deform my hand to the smallest diameter I can achieve) it has to be at least a diameter of $d_h=80 text mm$.

Let's take the 80 mm giant finger, that sounds reasonable, right? So to fit me, the giant has fingers of a diameter that is scaled up by a factor of $f=frac d_hd_rf=frac 8022=3.63$. Total height would be 7.26 meters or 24 feet, 5.7 inches⁽¹⁾. In either case pretty gigantic... but is that reasonable?

^{(1) - If the giant wore the ring on the thumb, the factor would be lower, at merely a $frac8025=3.2$, so just 6.4 meters/21'.}

Reality check time!

Picking out my trusty old matter on the Square-Law, and a BBC documentation we know that humans at 2.5 meters, we have critical health conditions. But if we blow up the bone diameters some, we could go somewhat bigger, but also reengineer other parts... Let's skip the heart and muscles and such.

Just from the bones alone, James Kakalios does offer the basics for an answer in his book The Physics of Superheroes, using Giant Man as an example in chapter 10. Does Size Matter? - The Cube-Square Law.

If Giant-Man is going to maintain a constant density as he
grows, then his mass must increase at the same rate as does his
volume

And we know, Volume goes by length to the cube. A Giant 3.5 times our height occupies approximately 12.25 times the area and has 42.875 times our volume - and thus weight⁽²⁾. But why do we want to know the area? Actually, we want to know the cross section of some bones, as we know rather well how well bones hold up stress depending on the cross section:

The compressive strength of an object, such as the femur in your thigh or the vertebrae in your spine, is determined by its cross-sectional area—that
is, the area of one of its faces if it were a rectangular solid.

Suppose at his normal height Dr. Pym is six feet tall and weighs
185 pounds [m_h]. His femur at his normal height can support a weight
of 18,000 pounds while a single vertebra [$m_s_v$] can support 800 pounds

So, That's an easy thing to calculate, right? Well yes! Let's assume our giant is Hank Pym. His factor is, as we calculated above, $f=3.63$.

So, $m_g=m_h text lbstimes f^3=8848text lbs$ while $m_s_vg=m_s_v text lbstimes f^2=10541text lbs$. You see easily: Our human vertebra shape scaled up can still support the weight of our giant! Oh, and our giant has just roundabout 1700 pounds of force as a safety margin: At the force of 119% of the own weight the giant's vertebra snaps, while a human has a safety factor of 432%.

As we see, our Giant is still able to live with just about a human skeleton scaled up, but they have a very fatal weakness of strikes to the head and tripping: falling flat induces forces well above the own body's mass. Slipping on a wet surface and not catching yourself is for humans not usually deadly, but for our 3.6 times scaled giants that would be rather deadly.

The muscles and such surely can be addressed rather easily, making them more heavyset and changing their composition. Even increasing all the muscles by an extra factor could mitigate some problems of the biological side (blood flow etc). But that surprisingly does not change the geometry of fingers a lot: A finger contains only a minimal amount of muscles, most o the force in a finger is generated in the lower arm and transmitted via ligaments.

^{(2) - Note that I speak about weight not mass: Weight is a force, which is on earth linearly related to the mass of an objects: $vec F=mvec g$ has a static factor $g=9.81fractext mtext s ^2$ directed towards the center of the planet, so for simplicity's sake, it boils down to this: An object's Weight is the Force created by its Mass times 10, directed to the center of the planet.
This allows us to speak of Weight in terms of Kilograms while it should technically be noted in Newtons - the conversion factor $g=9.81$ (or the rounded $g=10$) is easy enough to keep on the sidelines though, as it rarely matters in a planetary environment.}

Vulnerability

As to the vulnerability to head injuries and tripping, if the giant had much thicker cranial bones and was much stockier/more robust (while still being recognisably human in proportion), would you think this would solve that issue somewhat? Are denser and stronger bones reasonable for such a being? – Arkenstein XII

It's not the skull that is the main problem (A skull 3.6 times our skull might arguably survive the impacts of most small arms fire - it's thicker than hip bones). The true problem is the vertebrae: if one doesn't scale up the diameter of the spine by an additional factor, the spine (and for the matter, other bones) would shatter under a normal fall, as that easily exceeds the small 120% of force it is made to keep up against, while humans falling often manage to buffer dangerous falls into their safety range of below 430% of their body weight.⁽²⁾

When falling, the body typically impacts with at least two (a stumble) to 10 times the force of its mass (being shoved backwards onto the edge of a pedestrian walkway). To retain the same safety factor our human bones have, the bones need to be not 3.63² times as thick as a humans (normal scaling) but 3.63³ times as thick. This though would flaw our premise of scaleability of the finger bones.

To some degree, it can be mitigated by denser bone material. Typical quotes for human bone range from 1 to 1.9 g/cm³^source. Think at least elephant bone density, not human bone density. Some Blainville's Beaked Whale's bones are quoted to be 2.6 g/cm³.

Osteometric parameters show that the relationship of the length of the femur to the circumference is 2.5, 2.75 and 2.8 in elephant, horse and cattle respectively. Similarly, humerus length to circumference is 2.3 in the three species showing isometric scaling. There is a positive allometric scaling be-tween bone weight and bone length; the ratio of femur length to weight is 205 g/cm, 72 g/cm and 64 g/cm in elephants, horses and cattle. The ratio of weight of the humerus to length or weights of the humerus plus femur to their combined length is a good estimate of the body weight in kg= $(fracwhlhtimes 10)$.^here

The rough gist of that paragraph is: The femur bones of elephants are about 3 times as dense per length as those of bovine cattle while having a slightly more slender shape. This means that they are in fact of a more dense design. Note that their notation is kilograms of force⁽²⁾.

tl;dr: Conclusion

The giants are about 3.2 to 3.6 times as tall as humans (depending on which finger it comes from) and their skeletons are rather similar to humans. Their muscles might be more suited for fast acceleration in contrast to low tiring as humans, which might make them more stocky and heavyset in look, somewhat ogrish. Remember though, that their stride is much larger, so they are still faster than humans, even though they would tire after a shorter time than humans. Their bones might be of a stronger makeup than humans to compensate the lower safety factor the cube-square-law bestows upon them.

Are giants magical?

If you have magic, then the giant is probably proportional. The ratio of finger-radius-to-arm-radius is going to be about the ratio of height-to-height. I'd be real dubious about that 10-to-1 number, though. You might want to check that with your own fingers. Eyeballing it on myself, I get something a lot closer to 4-to-1. Alternately, you might want a giant who's somewhat thicker and stockier all around (ogre/dwarf build) in which case you could reasonably trim the height down by about half.

If you haven't got magic, then you have a lot of things to figure out about stuff like how their limbs sustain their weight and whatnot, and "how do I get the finger and arm radius to match up" is the least of your problems. On the bright side, you can fix a bunch of the simplest squared/cubed issues by having it be a particularly cold planet.

I see that Trish answered while I was typing. I'll give my answer anyway because it doesn't require complicated formulae to understand.

Radius is a length. For scaling up distance you don't have to worry about the square-cube law.

Estimating finger diameter to forearm diameter (and equivalently radius).

Hold two fingers together. Are the as wide as your wrist? Probably not unless you have fat fingers and skinny wrists.

Try with three fingers. In my case that's not enough.

Try holding four fingers together. In my case that's about right, maybe a little over.

So for me, the ratio between fingers and wrist is about 4:1

Because we are only dealing with distance at this point, you just scale everything up by the same ratio.

Thus if the human is 6 ft tall, a giant of exactly the same proportions will be 4 x 6 = 24 ft tall.

Now we come to the weight

Think of it this way. Suppose you have a cube of wood that is 1 inch per side. It's volume is 1 x 1 x 1 = 1 cubic inches.

Now we make a bigger cube of these small cubes. If the big cube is 4 small cubes high, 4 wide and 4 deep, then the volume of the big cube is 4 x 4 x 4 = 64 cubic inches.

So by multiplying the height of the cube by 4, you have multiplied the volume (and therefore the weight) by 64.

So if the human weighs 140 lbs the giant will weigh 140 x 64 = 8,960 lbs even though only four times taller.

That difference in scaling between height and weight is what causes the problem.

EDIT

As Trish points out, the area of a cross-section of any equivalent part of the body goes up in proportion to the square. Thus if you chopped the human's and giant's arms off at the equivalent places, the area of the cross-section would be 4 x 4 = 16 times bigger for the giant. (Again this is assuming the giant is 4 times as tall as the human)

It's entirely reasonable for the giant to have proportionally thicker limbs. This will help him walk.

Consider a giant 2.5x the overall size of a human with 1.5x proportionally thicker limbs. He'll weigh roughly 2.5^3=15.6x what a human does, and have legs roughly (2.5×1.5)^2=14x thicker. A few handwaves about bone and muscle density and this should actually support him. Square-cube problems averted! (This is the only solution to these equations)

It also means that his finger radius will be 2.5×1.5=3.75x that of a human, just enough for his ring to be worn as an armband.

Why do giants who only need thicker legs also have thicker arms? Pleiotropy, aesthetics, evolutionary history, they crawl sometimes...

I think at 1.5x thicker limbs, the giant will still look human, just noticeably heavyset. You might want to get a 3d model and check that, though

If you are worried about scientific plausibility, make your hero a little boy who saves the giant. A normal man who saves a giant would be saving someone much stronger than himself anyway, so making the hero a little boy simply increases the fact that someone weaker saves someone stronger from a situation where strength is not enough.

And if you can't change the story enough to make the hero a little boy when he saves the giant, then remember this anyway and maybe have a child save a giant in another story sometime and be rewarded with something giant-sized, like a magical coronet that the kid uses as a hulu hoop.

So if you can make your kid about 8 to 12 and about 4 to 5 feet tall a giant that might be 3 to 4 times as tall as him might be about 12 to 20 feet tall and a bit more plausible than a giant 3 to 4 times as tall as a man 6 feet tall, and thus 18 to 24 feet tall.

This is especially important if your giant is not a nonhuman member of a different species, but supposed to be merely a very big human.

Another factor is variation in the width of human body parts. I happen to have thin wrists and small hands that aren't much wider than the wrists, so that I can squeeze my hands to only about 1.1 times the width of my wrists. But some men who are the same height as me have much bigger hands and thicker wrists and maybe thicker fingers and thumbs.

And if your giant is nonhuman he can have somewhat thicker fingers proportionally than a human.

PS I think I remember somewhere seeing solid objects made of some materials that are flexible enough to be pulled open and then allowed to close, so that if the giant's ring is made of such a material it could be pulled open and put around the hero's wrist and then released and allowed to snap closed around the wrist. I don't know how many times a metal ring could do that without getting metal fatigue and snapping, but probably both the giant and the smaller person intended to wear it for the rest of their lives when they put it on.

PPS I notice you say the hero saves the captured giant. Then either the hero is on the giant's side and sneaks into the enemy camp or fortress to do something like getting the keys and unlocking the giant's cell, or else the hero is a member of the enemy society, living in the enemy city or is present in the enemy armed forces, and maybe convinces or orders the others to make a deal with the giant and release him instead of burning the giant at the stake, or whatever.

And certainly there are many examples of children being present in military forces.