Buoyancy Calculator
Calculate buoyant force from displaced volume, fluid density, and gravity. A clear way to check whether an object will float or sink.
Buoyancy Calculator
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What the buoyancy calculator does
Drop something into water and the water pushes back up on it. This calculator works out that upward push, the buoyant force, from the volume of fluid the object displaces, the density of the fluid, and the strength of gravity. Alongside the force, it reports the mass of fluid that gets displaced.
Below is what buoyancy is, the principle behind it, the equation it uses, and a worked example.
How to use it
- Enter the displaced volume, the volume of fluid the object pushes aside, with its unit.
- Enter the fluid density (it defaults to seawater) and the gravity, which you can give in metres per second squared or in multiples of standard gravity.
- Press Calculate for the buoyant force and the displaced fluid mass, or Reset to clear it.
What buoyancy is
Buoyancy is the upward force a fluid exerts on anything placed in it. It is why a beach ball leaps back to the surface when you let go of it underwater, why a boat made of heavy steel rides on the sea, and why you feel lighter standing in a swimming pool. The fluid, whether liquid or gas, pushes up on whatever sits within it.
The force comes from pressure. In any fluid, pressure increases with depth, so the bottom of a submerged object sits in higher pressure than its top. The fluid therefore pushes up on the bottom harder than it pushes down on the top, and the difference is a net upward force. That net upward push is buoyancy, and this calculator gives its size.
Archimedes' principle
More than two thousand years ago, Archimedes worked out the elegant rule that governs this force, and it still carries his name. Archimedes' principle says that the upward buoyant force on an object equals the weight of the fluid it displaces, that is, the weight of the fluid that would have filled the space the object now occupies.
This is a remarkably tidy result. To find the upward push, you do not need to track the pressure over every surface of a complicated shape. You only need to ask how much fluid the object shoves out of the way, and weigh that. An object that displaces a litre of water feels an upward force equal to the weight of a litre of water, no matter what the object is made of or what shape it takes. That is the principle this calculator is built on.
The equation it uses
Written as a formula, Archimedes' principle gives the buoyant force F from the fluid density ρ, the strength of gravity g, and the displaced volume V:
F = ρ × g × V
The density times the volume is just the mass of fluid displaced, and multiplying by gravity turns that mass into a weight, which is the buoyant force. The calculator reports that displaced fluid mass too, as ρ times V, since it is a useful quantity in its own right. A denser fluid, or a larger displaced volume, gives a stronger upward push, which is why salty seawater buoys you up more than fresh water and why a bigger hull can carry a heavier load.
Why things float or sink
Whether an object floats comes down to a contest between two forces: the buoyant force pushing up and the object's own weight pulling down. If the upward buoyant force can match the object's weight, the object floats; if it cannot, the object sinks. That is the whole story, and it explains some things that seem puzzling at first.
A solid lump of steel sinks because the weight of water it displaces is far less than its own weight. Shape that same steel into a hull, though, and it encloses a large volume, displacing far more water before it submerges, enough that the buoyant force can equal the ship's weight. The ship floats not because steel is light, but because its shape displaces a great deal of water. The same logic lifts a hot-air balloon, which displaces a huge volume of air, and lets a hydrometer float at a depth that reveals a liquid's density. Comparing buoyant force to weight tells you which way an object will go.
Units and precision
The calculator works in SI units underneath, taking the volume in cubic metres, the density in kilograms per cubic metre, and gravity in metres per second squared, while the menus let you enter values in many other units. The density defaults to about 1,024 kilograms per cubic metre, a typical value for seawater, and gravity can be set in multiples of standard gravity, where one unit is the familiar 9.80665 metres per second squared at Earth's surface. The buoyant force comes out in newtons and the displaced mass in kilograms. The relationship is exact for a fully known displaced volume. Results carry several decimal places.
A worked example
Suppose an object displaces 1 cubic metre of seawater, with seawater's density of about 1,024 kilograms per cubic metre and gravity at its standard value.
The displaced fluid mass is ρ × V = 1,024 × 1 = 1,024 kilograms, and the buoyant force is F = ρgV = 1,024 × 9.80665 × 1 ≈ 10,040 newtons pushing upward. If the object itself weighs less than that, it floats; if more, it sinks. A cubic metre is a lot of displacement, which is why large hulls can support so much weight.
Questions people ask
How do you calculate buoyant force?
Multiply the fluid density by gravity and by the displaced volume, F = ρgV. This equals the weight of the displaced fluid, which is Archimedes' principle.
What is Archimedes' principle?
It states that the upward buoyant force on an object equals the weight of the fluid it displaces. You find the force by weighing the fluid the object pushes aside, whatever the object's shape or material.
Why does a heavy steel ship float?
Because its hull encloses a large volume and displaces a great deal of water before submerging. The buoyant force from that displaced water equals the ship's weight, so it floats despite steel being dense.
Why is it easier to float in the sea than in a pool?
Seawater is denser than fresh water, so for the same displaced volume it gives a larger buoyant force. The denser fluid pushes up harder, making you more buoyant.
References
A quick note on where the physics comes from. Buoyancy and Archimedes' principle are standard fluid mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The SI units follow the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.
- OpenStax, University Physics Volume 1, Section 14.4, Archimedes' Principle and Buoyancy. https://openstax.org/books/university-physics-volume-1/pages/14-4-archimedes-principle-and-buoyancy
- HyperPhysics, Buoyancy. http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html
- National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811
Bibek Lal Karna is a PhD student and graduate teaching assistant at the University of Mississippi, with deep interests in theoretical and gravitational physics. He is also the founder of NRCC and is strongly engaged in scientific teaching and communication. At Eon Tools, he reviews physics tools.