Archimedes’ principle states that any object placed in a fluid is pushed upward by a buoyant force equal to the weight of the fluid the object displaces. The force is found with F_b = ρVg, where ρ is the fluid’s density, V the displaced volume, and g gravity. An object floats when this upward force balances its weight.
Drop a steel spanner into a sink and it plunges straight to the bottom. Yet a steel ship the length of three football pitches sits calmly on the ocean, carrying thousands of cars. Same metal, opposite fate. The rule that explains both — and the floating ice in your drink, and the lift under a hot-air balloon — is one of the oldest in physics.
It came from a puzzle about a king’s crown more than two thousand years ago, and it still decides how submarines dive, how life jackets save lives, and how much of every iceberg hides beneath the waves. Once you can picture the fluid being pushed out of the way, the whole thing clicks.
What Is Archimedes’ Principle?
Picture lowering a sealed box into a bathtub. The water has to go somewhere, so it rises — the box has displaced a certain volume of water. Archimedes’ principle says the fluid pushes back: it shoves the box upward with a force exactly equal to the weight of the water that was moved aside.
That upward push is the buoyant force (also called upthrust). It does not care what the object is made of, only how much fluid the object displaces and how heavy that fluid is. This idea is often written simply as the Archimedes principle, and it applies to every fluid — water, oil, even the air around you.
So whether an object floats or sinks is a contest between two forces: gravity pulling it down by its weight, and buoyancy pushing it up by the weight of displaced fluid. Win the contest and it rises; lose and it sinks; tie and it hovers in place.
The man behind it, Archimedes of Syracuse (c. 287–212 BC), reportedly hit on the idea in his bath and shouted “Eureka!” The famous story has him testing a king’s gold crown for cheating. Whether he literally used a bathtub is debated by historians — a careful hydrostatic balance is the more likely method — but the physics he uncovered is rock-solid.
The Buoyant Force Formula
The size of the upthrust comes straight from the weight of displaced fluid. Since weight is mass times gravity, and the displaced mass is the fluid’s density times the displaced volume, the buoyant force is:
Every symbol has a job, and each one carries an SI unit. Get the units right and the answer lands in newtons every time.
| Symbol | Quantity | SI unit |
|---|---|---|
| F_b | Buoyant force (upthrust) | newton (N) |
| ρ (rho) | Density of the fluid | kg/m³ |
| V | Volume of fluid displaced | m³ |
| g | Gravitational field strength | ≈ 9.81 m/s² |
One trap worth flagging now: ρ is the density of the fluid, never the object. Swap in the object’s density by mistake and the physics falls apart.
Apparent weight and the float fraction
When an object hangs submerged on a scale, the scale reads less than its true weight, because buoyancy lifts part of the load. That reduced reading is the apparent weight:
For something that floats, there is a neat shortcut for how deep it rides. The fraction of its volume sitting below the surface equals the ratio of the two densities:
This single ratio is why a denser object floats lower, and why ice — slightly less dense than water — barely pokes above the surface. You can run the numbers for any object and fluid with our Buoyancy Calculator.
How Buoyancy Works
Where does the upward push actually come from? Pressure. In any fluid, pressure grows with depth, because deeper layers carry the weight of everything above them. The relationship is P = ρgh — go deeper, and h grows, so the pressure climbs.
Now think about a submerged block. The fluid presses on every face, but the bottom face sits deeper than the top face. So the upward push on the bottom beats the downward push on the top. That difference is the buoyant force.
Put numbers on it. If the top face is at depth d and the block has height h and face area A, the bottom sits at depth d + h. The net upward force is the bottom force minus the top force:
F_b = ρg(d + h)A − ρg(d)A = ρg(hA) = ρgV. The depth d cancels, leaving exactly ρVg — the displaced volume V = hA times ρg. NASA’s Glenn Research Center walks through this same pressure-difference derivation in its Archimedes’ principle activity.
Fluid pressure is greater on the deeper bottom face than on the top, and that imbalance produces the net upward buoyant force.
Try it yourself. In the lab below, change the object’s density, its volume, and the fluid. Watch the weight, buoyant force, apparent weight and submerged percentage update live as the block settles to float or sink.
Real-World Examples of Buoyancy
Archimedes’ principle is not a textbook curiosity — it is doing quiet work all around you. Here are five places it shows up.
1. Why steel ships float
A solid lump of steel sinks because steel is far denser than water. A ship dodges this by being mostly hollow. Its hull encloses a huge volume of air, so the average density of the whole vessel — steel plus air — drops below that of water, and it floats. Overload it, and the average density climbs until it sinks.
2. How submarines dive and surface
Submarines play with their own weight on purpose. To dive, they flood ballast tanks with seawater, raising their average density above the surrounding water. To surface, compressed air blows that water back out, the average density falls, and buoyancy lifts them up.
3. Hot-air and helium balloons
Here the fluid is air, not water. A balloon rises when it displaces enough air to be lifted — that is, when its average density falls below the air around it. Helium does this because it is far lighter than air; a hot-air balloon does it by heating the air inside until it thins out.
4. Hydrometers
A hydrometer is a weighted float used to measure a liquid’s density. Drop it into a fluid and it sinks until it displaces its own weight. In a denser liquid it sits higher; in a thinner one it sinks lower — and the depth is read straight off a scale.
5. Icebergs
Ice is only slightly less dense than seawater, so an iceberg floats with just a sliver above the surface. Plug the densities into the float-fraction rule and the famous “tip of the iceberg” turns into a hard number — about 89.5% stays hidden underwater.
An iceberg floats with roughly nine-tenths of its volume below the surface — the ratio of ice density to seawater density.
Common Misconceptions About Buoyancy
Buoyancy is one of those topics where intuition leads people astray. Four wrong beliefs come up again and again.
“The buoyant force depends on the object’s weight”
It does not. Buoyancy depends only on the fluid’s density, the displaced volume, and gravity — the object’s own weight and material never enter F_b = ρVg. As Georgia State University’s HyperPhysics notes, equal volumes of cork, aluminium and lead, fully submerged, all feel the same buoyant force. What differs is their weight, which decides whether they float.
“Heavy things always sink”
Weight alone settles nothing — average density does. A 100,000-tonne ship floats while a 5-gram nail sinks, because the ship’s average density (steel plus enclosed air) is below water while the nail’s is not. Compare the object’s average density with the fluid’s, never its raw mass.
“The buoyant force grows as the object sinks deeper”
For a rigid object in an incompressible fluid like water, it does not. Pressure rises with depth on both the top and bottom faces, but their difference stays the same, so F_b = ρVg is unchanged at any depth. (Gas-filled objects are the exception — they compress, shrinking V, and so lose buoyancy as they descend.)
“A sunken object displaces its own weight of fluid”
Only a floating object does that. A fully submerged object displaces its own volume of fluid, and the buoyant force equals the weight of just that displaced fluid — which, for anything that sinks, is less than the object’s own weight. Mixing up “displaces its weight” and “displaces its volume” is the single most common slip.
How Buoyancy Relates to Density, Pressure and Weight
Buoyancy never acts alone. It is really a story about density set against the fluid, pressure that builds with depth, and the weight it works against.
Density is the decider. If an object’s average density is below the fluid’s, it floats; above, it sinks; equal, it hovers. The table below shows where common materials land in fresh water.
| Material / fluid | Density (kg/m³) | In fresh water (1000 kg/m³)? |
|---|---|---|
| Helium (gas) | 0.18 | Rises (in air) |
| Air | 1.225 | — |
| Cork | ≈ 240 | Floats |
| Ice | 917 | Floats |
| Vegetable oil | ≈ 920 | Floats |
| Fresh water | 1000 | Reference |
| Seawater | 1025 | Sinks slightly |
| Aluminium | 2700 | Sinks |
| Iron | 7870 | Sinks |
| Lead | 11,340 | Sinks |
| Mercury (liquid) | 13,534 | Sinks |
| Gold | 19,300 | Sinks |
Pressure is the engine. The buoyant force exists only because fluid pressure climbs with depth, pushing harder on the bottom of an object than the top. This is the same depth-pressure idea that governs how fast things fall through a fluid, where buoyancy and drag both resist gravity — see our guide to terminal velocity.
Weight is the opponent. Floating is simply the balance point where the upthrust equals the object’s weight — a state of equilibrium, exactly the kind of force balance covered in Newton’s laws of motion. When the forces do not balance, the object accelerates, and the net force follows directly from Newton’s second law.
