Buoyancy Calculator — Archimedes' Principle, Float or Sink, and Displaced Volume
According to Archimedes' principle, the buoyant force on an object equals the weight of the fluid it displaces. The float-or-sink decision comes from comparing object density with fluid density: less dense floats, more dense sinks, and equal density is neutrally buoyant.
This calculator supports fully submerged and partially submerged cases for ship draft, diver ballast, buoy design, and iceberg visibility. Use preset fluids such as water, oil, alcohol, mercury, and density of seawater, or pair it with the material density calculator for object density lookup.
Calculation mode
Object section
Pine: 500 kg/m³
Shape selector
Rectangular block dimensions
V = L × W × H
Fluid section
Fresh water (20°C): 998.2 kg/m³ — Standard lab / pool
Submersion
FLOATS
Object density (500 kg/m³) < Fluid density (998.2 kg/m³)
Net force at selected submersion: +4.8873 N
Forces
Volumes
Masses
Densities
Calculation
F_b = ρ_fluid × V_submerged × g
= 998.2 × 0.001 × 9.81
= 9.7923 N
Archimedes' Principle
Archimedes' principle states that any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This principle was reportedly discovered by the ancient Greek mathematician Archimedes of Syracuse (c. 287–212 BC), who is said to have leapt from his bath shouting "Eureka!" upon realising it could be used to determine the purity of a gold crown without melting it. For the underlying quantity, see what is density.
Where:
- F_b = buoyant force (N)
- ρ_fluid = density of the fluid (kg/m³)
- V_submerged = volume of the object submerged in the fluid (m³)
- g = gravitational acceleration (9.81 m/s²)
The net force on the object is:
The float-or-sink outcome depends entirely on the comparison between object density and fluid density:
- If ρ_obj < ρ_fluid: net force is upward → object floats
- If ρ_obj > ρ_fluid: net force is downward → object sinks
- If ρ_obj = ρ_fluid: net force is zero → object is neutrally buoyant
Note that shape does not affect whether a uniform solid object floats or sinks — only the average density matters. A solid steel cube and a solid steel sphere of the same mass both sink, regardless of shape. However, a hollow steel sphere with enough air inside has a lower average density than water and floats — this is the principle behind ships.
A steel ship floats not because steel floats, but because the ship's hull encloses a large volume of air, making the average density of the entire ship (steel + air + cargo) less than water. The ship sinks until the weight of displaced water equals the total weight of the ship. This equilibrium point determines the ship's draft (depth below waterline). As cargo is loaded, the ship sinks lower; as cargo is unloaded or fuel is consumed, it rises.
Partial Submersion: The Floating Fraction Rule
When a floating object reaches equilibrium, the buoyant force exactly equals its weight. This means the weight of displaced fluid equals the weight of the object. From this condition, we can derive the fraction of the object's volume that is submerged:
This simple ratio explains one of the most famous facts in oceanography: approximately 89% of an iceberg is below the waterline. density of ice: 917 kg/m³. Seawater density: 1,025 kg/m³. Submerged fraction = 917 / 1,025 = 0.895 = 89.5%. Only 10.5% of an iceberg is visible above the surface — the origin of the phrase "tip of the iceberg."
The same rule applies to any floating object:
- Pine wood (500 kg/m³) in fresh water (1,000 kg/m³): submerged fraction = 500/1,000 = 50%. Half above, half below.
- Olive oil (909 kg/m³) floating on water (998 kg/m³): submerged fraction = 909/998 = 91%. The oil layer is mostly below the water surface level — only 9% protrudes above.
- A person (average ~985 kg/m³) in fresh water (1,000 kg/m³): submerged fraction = 985/1,000 = 98.5%. Almost entirely submerged, which is why swimming requires active effort to keep the head above water.
Buoyancy in Different Fluids
The same object can experience very different buoyant force in different fluids. The table below uses a 1 kg object with volume 0.001 m³ (density 1,000 kg/m³). For reference values, compare density of water and density of oil.
| Fluid | Fluid Density (kg/m³) | Buoyant Force (N) | Net Force (N) | Result |
|---|---|---|---|---|
| Air | 1.225 | 0.012 | −9.798 | Sinks |
| Gasoline | 748 | 7.34 | −2.47 | Sinks |
| Ethanol | 789 | 7.74 | −2.07 | Sinks |
| Fresh water (20°C) | 998 | 9.79 | −0.01 | Nearly neutral |
| Fresh water (4°C) | 1,000 | 9.81 | 0.00 | Neutral |
| Seawater (avg) | 1,025 | 10.06 | +0.25 | Floats |
| Whole milk | 1,030 | 10.10 | +0.29 | Floats |
| Dead Sea water | 1,240 | 12.16 | +2.35 | Floats (strongly) |
| Glycerol | 1,261 | 12.37 | +2.56 | Floats (strongly) |
| Mercury | 13,534 | 132.8 | +123.0 | Floats (very strongly) |
Object: mass = 1 kg, volume = 0.001 m³, density = 1,000 kg/m³. Buoyant force = ρ_fluid × 0.001 × 9.81. Net force = F_b − W = F_b − (1 × 9.81). Positive net force = upward (floats); negative = downward (sinks). In mercury, even a solid lead block (11,340 kg/m³) floats — mercury is so dense that it supports almost any common material.
Real-World Applications
Ship Design and Loading
Naval architects use Archimedes' principle to calculate a ship's displacement — the total weight of water displaced when the ship is at its design draft. A ship is in equilibrium when displacement equals total ship weight (hull + machinery + fuel + cargo + crew). The Plimsoll line marks the maximum allowable draft in different water densities (fresh vs salt) and seasons (summer vs winter). Every tonne of cargo added sinks the ship by a calculable amount called the "tonnes per centimetre immersion" (TPC).
Scuba Diving and Buoyancy Control
Scuba divers achieve neutral buoyancy by adjusting the air volume in their buoyancy compensator device (BCD). A diver with wetsuit, tank, and equipment has a combined density close to water. Adding air to the BCD increases volume (decreasing average density) for ascent; releasing air decreases volume (increasing average density) for descent. Divers also use lead weights to offset the buoyancy of the wetsuit (neoprene foam, density ~200 kg/m³, is very buoyant). The ballast weight calculator above is directly applicable to dive planning.
Submarine Ballast Tanks
Submarines control depth by flooding or emptying ballast tanks with seawater. When tanks are full of water, the submarine's average density exceeds seawater density and it sinks. When compressed air blows water out of the tanks, average density drops below seawater and the submarine rises. At neutral buoyancy, the submarine hovers at a fixed depth with no power required for depth control — only for forward propulsion.
Hydrometer and Density Measurement
A hydrometer is a simple float instrument that measures fluid density by how deep it sinks. It is calibrated so that the waterline on the stem indicates density directly. Hydrometers are used to measure battery acid concentration, antifreeze strength, alcohol content in brewing, and milk quality. The instrument is a direct physical implementation of Archimedes' principle: denser fluid → more buoyant force → hydrometer floats higher. For related ratio work, use the specific gravity calculator.
Related Calculators
density to weight calculator
Calculate the mass and gravitational force (weight in N/kN) of any object by material and shape. Includes 8 shape templates and Moon/Mars gravity presets.
liquid density calculator
Find the density of water, seawater, oils, and 60+ other liquids at any temperature. Essential for accurate buoyancy calculations in non-standard fluids.
density of seawater
How salinity, temperature, and depth affect ocean water density — critical for marine engineering and diving buoyancy calculations.
Frequently Asked Questions
How do I calculate buoyant force?
Use the formula F_b = ρ_fluid × V_submerged × g. Multiply the fluid density (kg/m³) by the submerged volume of the object (m³) by gravitational acceleration (9.81 m/s²). The result is in Newtons. For example, a 2-litre object (0.002 m³) fully submerged in fresh water (998 kg/m³): F_b = 998 × 0.002 × 9.81 = 19.58 N.
Does shape affect whether an object floats or sinks?
For a uniform solid object, no — only average density matters. A solid steel cube and a solid steel sphere both sink because steel (7,850 kg/m³) is denser than water (998 kg/m³), regardless of shape. However, a hollow steel object (like a ship) can float if the enclosed air makes the average density of the entire object less than water. Shape affects stability (how the object orients itself) but not the fundamental float-or-sink outcome for uniform solids.
What fraction of an iceberg is underwater?
Approximately 89.5%. Ice density is 917 kg/m³ and seawater density is approximately 1,025 kg/m³. The submerged fraction = 917 / 1,025 = 0.895, meaning 89.5% of the iceberg is below the waterline and only 10.5% is visible above the surface.
Why is it easier to float in the Dead Sea?
The Dead Sea has an extremely high salinity (approximately 280–340 g/kg), giving it a density of about 1,240 kg/m³ — far higher than the human body's average density of approximately 985 kg/m³. The submerged fraction of a person in the Dead Sea is only 985/1,240 = 79%, meaning 21% of the body floats above the surface. In fresh water, 98.5% of the body is submerged, requiring active swimming to keep the head above water. For ocean reference values, see density of seawater.
How much ballast does a scuba diver need?
It depends on the diver's total volume (body + wetsuit + equipment) and the water density. A typical recreational diver in a 5 mm wetsuit in seawater needs approximately 6–10% of body weight in lead ballast to achieve neutral buoyancy. Use the Ballast Weight mode above: enter the diver's total mass and estimated volume, select seawater as the fluid, and the calculator outputs the required ballast mass.
Can anything float in mercury?
Yes — mercury (13,534 kg/m³) is so dense that most common materials float in it. Lead (11,340 kg/m³), which sinks in water, floats in mercury. Even gold (19,320 kg/m³) sinks in mercury, but steel, aluminum, wood, and most plastics all float. A solid iron cannonball floats in mercury — a famous demonstration in physics classrooms.