1. The square-cube law
Scale every length of a body by a factor s. Areas grow as s², volumes (and so mass) as s³. A bone, a muscle or a tendon carries load in proportion to its cross-section, but the weight it must carry grows with the volume. Stress is force over area, so for a pure scale-up
stress σ = F / A ∝ s³ / s² = s
A troll twice as tall has bones that are four times stronger but a body eight times heavier: every bone works twice as hard. This is why you cannot just draw a human 4 m tall and expect it to work, and why real large animals (elephants, sauropods) have very different proportions from small ones.
2. The thigh bone model
The femur shaft is a hollow tube of compact bone: outer diameter 28 mm, inner 15 mm (area 4.4 cm², second moment of area I = π(D⁴ − d⁴)/64). The load acts a little off the shaft axis (offset e = 20 mm for a human leg), so the bone is squeezed and bent:
σ = F/A + F·e·(D/2)/I F = n · m · g
n = 0.6 (stand), 3 (walk), 5 (run), 8 (landing) body weights
The numbers n are peak hip-joint forces from gait studies (they include the pull of the muscles around the hip). With σ_u = 150 MPa compressive strength the model gives a human a safety factor of about 6 when walking, 3.5 when running, and 2.2 when landing from a jump, which matches the "factor of 2 to 4 at peak effort" reported for real limb bones. A second failure mode is Euler buckling, P_cr = π²·E·I / L² with E = 17 GPa. Safety factor = the smaller of strength/stress and P_cr/F.
With a pure scale-up (p = 1) every term follows a power law and the height at which the safety factor reaches 1 has a closed form, s_max = σ_u / σ(s=1). The sliders let the bones grow thicker than isometry, D ∝ s^p (exponent p, real mammals use about 1.1), and straighten the leg (smaller load offset e), which is what elephants do. Body mass then follows m = m₀·s³·[(1 − λ) + λ·s^(2p−2)] with 45 % of the body (the limbs) thickening.
3. Muscles run out first
A muscle's force is set by its cross-section (about 0.3 MPa of effective tension), so the strongest push one leg can make grows as s^(2p), again one power of s slower than the weight. The model gives a human leg a maximum single-leg push of 4 body weights; walking needs 1.2, running 2.5, landing 4 (with the eccentric bonus of 1.5×). Effort = demand / capacity. For a human-shaped troll walking effort passes 100 % near 5.8 m, long before the thigh bone breaks, which is also why the tallest real people struggle to walk and why a straight, columnar leg (less muscle needed to hold the joint) helps so much.
4. Walking, speed and food
Walking is a pendulum, so with gravity g the natural speed is v ∝ √(g·L) (constant Froude number) and the stride frequency falls as 1/√s. A giant walks faster but takes slower steps. The metabolic cost of walking is taken as 2.3 J per kg per metre (constant per kilogram by geometric similarity; the option "animal trend" uses M^−0.3 from running animals). Resting metabolism follows Kleiber's law P = 3.39·M^0.75 W, so food needed per day grows slower than mass but faster than the stomach's surface can easily supply. Bread is taken as 2500 kcal/kg.
5. Heat
Heat leaves through the skin (area ∝ s²) but is made in the whole body, so walking heat per square metre of skin grows as s^1.5. Skin heat loss uses clothing insulation U = 1/(0.155·clo + 0.11) and a skin-to-air difference of (32 °C − air). Sweating can remove at most about 365 W per m² of skin. If walking heat is more than loss plus sweat, the body stores heat at net / (m·c) with c = 3470 J/(kg·K); the "+2 K" time is how long until a heat-stroke warning level.
6. Blood pressure and the heart
To keep blood flowing to the brain the heart must beat the hydrostatic column to the head: ΔP = ρ·g·h. We require brain mean arterial pressure of 70 mmHg (the human value), so
MAP_heart = 70 mmHg + ρ_blood · g · h_head, h_head = 0.30 m · s
For a giraffe (brain 2.0 m above the heart) this gives 226 mmHg; measured mean pressures are about 200 mmHg, a 13 % difference, in line with the model simplicity. At the ankle a standing human has 100 mmHg of hydrostatic pressure; a 4 m troll 230 mmHg, which would push fluid out of the capillaries (swollen legs) unless the skin is very tight, like a giraffe's. Heart rate is scaled as M^−0.25, resting cardiac output as M^0.75, and the left-ventricle wall thickness needed to keep the same wall stress follows Laplace's law (t ∝ P·r).
7. Falls, jumps and feet
Tripping from standing drops the centre of mass by about half the height: E = m·g·H/2, which grows as s⁴, while the energy a bone can absorb before breaking grows only as its volume (s^(2p+1), 3 for isometry). The "fall damage index" is the ratio, so a 4 m troll's fall is 2.3 times more dangerous to its bones than a human's, per fall. Jump height depends on the work a kilogram of muscle can do (about 4.4 J per kg), not on size, so h = w/g ≈ 0.45 m for everything from a flea to a human (to first order); a giant cannot jump over its own height. Feet are modelled as contact area A_f ∝ s^(1+p), so standing foot pressure grows as s for p = 1.
What is simplified
One bone (the femur) stands for the whole skeleton; real failure also happens at the spine, hip, knee and ankle, in tendons and in cartilage. Loads are peak values for a steady gait, bone strength is a single average value (it also varies with age and loading rate), cortical thickness is scaled with the outer diameter, and muscle capacity is a single human-based number. Metabolic, heat and blood-pressure rules are textbook allometry and meant to show trends, not to design a giant. The troll is a cartoon: no real skeleton was modelled, and the femur shown is the actual-size shape used for the numbers. Giants in stories also cheat with magic; the square-cube law does not.