Water Displacement Calculator
The Legend
King Hiero II of Syracuse commissioned a golden crown but suspected the goldsmith of fraud—mixing silver with the gold. He asked Archimedes to determine if the crown was pure gold without damaging it. While bathing, Archimedes noticed water spilling over the tub's edge as he entered. He realized he could measure the crown's volume by water displacement and compare its density to pure gold. Shouting "Eureka!" (I have found it!), he reportedly ran naked through the streets in excitement.
The Mathematics
Density: ρ = mass/volume (g/cm³)
Pure gold: ρ ≈ 19.3 g/cm³
Silver: ρ ≈ 10.5 g/cm³
Water: ρ ≈ 1.0 g/cm³
Buoyant Force: F_b = ρ_fluid × V × g
The upward force equals the weight of displaced fluid.
Apparent Weight: W_apparent = W_actual - F_b
An object feels lighter in water due to buoyancy.
Test: If the crown's density differs from 19.3 g/cm³, it's not pure gold. A crown of 1000g should have volume ≈ 51.8 cm³ if pure gold.