Archimedes' Principle Calculator
Enter the object volume and fluid type to find the buoyant force, apparent weight, and whether the object floats or sinks. Switch between metric and imperial units at any time, choose from eight common fluid presets, or enter a custom fluid density. The show-your-work panel traces every step of the calculation.
Formula
Worked example
A solid aluminum sphere with volume 0.001 m3 (1 litre) is fully submerged in fresh water (rho = 1000 kg/m3). Displaced mass = 1000 x 0.001 = 1 kg. Buoyant force = 1 x 9.807 = 9.807 N. Aluminum density is 2700 kg/m3, so mass = 2.7 kg, true weight = 2.7 x 9.807 = 26.48 N. Apparent weight = 26.48 - 9.807 = 16.67 N. The sphere sinks (apparent weight > 0).
What is 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. Archimedes of Syracuse reportedly discovered this while stepping into a bath and noticing the water level rise, recognizing that the volume of water displaced equalled the volume of his body. The principle applies to liquids and gases alike and underlies the behaviour of ships, submarines, hot-air balloons, and hydrometers.
The buoyancy formula
The buoyant force is calculated as: F_B = rho x g x V, where rho (kg/m3) is the density of the fluid, g (m/s2) is gravitational acceleration, and V (m3) is the volume of fluid displaced by the submerged portion of the object. This equals the weight of the displaced fluid (F_B = m_displaced x g). In imperial units the same formula applies with consistent units: density in lb/ft3, volume in ft3, and gravity in ft/s2, giving force in lbf.
Float or sink: the key comparison
Whether an object floats depends on comparing its average density to the fluid's density - not its absolute weight. If the object's average density is less than the fluid's, the buoyant force when fully submerged exceeds the object's weight, so the object rises and floats at a level where the displaced fluid weight exactly equals the object's weight. If the object is denser than the fluid, it sinks. This is why a solid steel ball sinks in water but a hollow steel ship floats: the average density of the ship (steel + enclosed air) is less than water.
Apparent weight and the lost weight method
When an object is submerged, it appears lighter by exactly the buoyant force. This apparent weight (W_apparent = W_true - F_B) is what a scale reads when an object hangs from it into a fluid. The 'lost weight' (F_B = W_true - W_apparent) lets you measure the buoyant force by simply weighing the object in air and then in the fluid. Dividing the lost weight by the fluid density and gravity gives the volume of the object - a classic method used in metallurgy and gemology to identify alloys by density.
Real-world applications
Ships and submarines: a ship's hull is shaped so that the average density of the vessel plus its contents is less than water. Submarines control buoyancy by pumping water in or out of ballast tanks. Hot-air balloons: heated air inside the envelope is less dense than the cool air outside, generating enough buoyant force to lift the balloon and its payload. Density measurement: by weighing a sample in air and in a known fluid (often water), the sample's density can be calculated precisely - jewellers use this to check gold purity. Hydrometers measure fluid density directly by floating at a depth set by buoyancy. The Dead Sea is so salty (density ~1240 kg/m3) that humans float effortlessly because a small displaced volume already outweighs a typical person.
Common fluid densities
| Fluid | Density (kg/m³) | Notes |
|---|---|---|
| Water (fresh) | 1000 | Standard reference |
| Sea water | 1025 | Average ocean salinity ~3.5% |
| Alcohol (ethanol) | 789 | Pure ethyl alcohol |
| Alcohol (methanol) | 791 | Methyl alcohol |
| Gasoline | 720 | Typical pump fuel |
| Diesel / fuel oil | 850 | Automotive diesel |
| Honey | 1420 | Variable by moisture content |
| Mercury | 13534 | Liquid at room temperature |
Reference densities at approximately 20 degrees C and standard pressure.
Frequently asked questions
What is Archimedes' principle in simple terms?
When you push an object into a fluid, the fluid pushes back with a force equal to the weight of the fluid you just displaced. If that upward push is bigger than the object's own weight, the object floats. If the object is heavier, it sinks.
How do I calculate buoyant force?
Multiply the fluid density (kg/m3) by the submerged volume (m3) and by gravitational acceleration (9.80665 m/s2). The result is the buoyant force in Newtons. For example, a 0.001 m3 object fully submerged in fresh water displaces 1 kg of water, giving a buoyant force of 1 x 9.80665 = 9.807 N.
Does the shape of the object affect buoyancy?
No - only the displaced volume matters, not the shape. A 1 kg steel sphere and a 1 kg steel cube of identical volume displace exactly the same amount of fluid and experience the same buoyant force. Shape does affect drag and stability, but not the buoyant force itself.
Why does a steel ship float when solid steel sinks?
A solid steel bar sinks because its average density (about 7800 kg/m3) far exceeds water (1000 kg/m3). A ship is hollow - it encloses a large volume of air. The combined average density of all the steel plus all the enclosed air is less than 1000 kg/m3, so the ship floats. The hull shape distributes the weight over a very large water-displacing volume.
Does buoyancy work the same on other planets?
Yes, the principle works everywhere a fluid exists. The buoyant force scales with local gravity, but so does the object's weight - they cancel out in the float/sink comparison. An object that floats on Earth floats on Mars too, though the absolute forces are smaller (Mars gravity is about 3.72 m/s2 vs 9.81 m/s2 on Earth). You can test this with the gravity field in this calculator.
What is apparent weight?
Apparent weight is the reading on a scale when an object is fully submerged in a fluid. It equals the object's true weight in air minus the buoyant force. Because the fluid pushes up, the object seems lighter. If the apparent weight is zero or negative, the buoyant force equals or exceeds the object's weight - it floats.