A steel ship weighs tens of thousands of tonnes, yet it floats. A solid steel bolt of the same metal sinks instantly in the same water. The metal is identical, so the difference cannot be the material. It is the shape — specifically, how much water each object pushes out of the way. The ship is mostly hollow and shoves aside a huge volume of water; the bolt displaces only its own small volume. That single observation is Archimedes' principle, and it governs everything from submarines to hot-air balloons.
This article explains where the buoyant force comes from, how to calculate it, and how to decide whether an object will float, sink, or hover. The arithmetic is simple; the place people slip is in deciding which volume to use.
Why this calculation matters
Buoyancy is the deciding force in any system where a body sits in a fluid. Naval architects use it to set how deep a hull rides and how much cargo it can carry. Submarine designers tune it deliberately, flooding and emptying ballast tanks to dive and surface. It explains why a hot-air balloon climbs, why a hydrometer measures the density of a liquid, and why a diver carries weights to stop bobbing back to the surface.
Get the buoyancy wrong and the consequences are not subtle. Overload a vessel and it rides too low, loses freeboard, and is at risk in rough water. Misjudge the buoyancy of a subsea structure during installation and it can refuse to sink, or sink too fast. Because the buoyant force depends only on displaced volume and fluid density, a quick calculation tells you immediately whether a design is workable before any metal is cut.
The core formula
Archimedes' principle states that the upward buoyant force on a body in a fluid equals the weight of the fluid that the body displaces:
F_b = rho_fluid * g * V_displaced
Here rho_fluid is the density of the fluid in kilograms per cubic metre, g is gravitational acceleration (about 9.81 m/s^2), and V_displaced is the volume of fluid pushed aside, in cubic metres. The force comes out in newtons.
The physical origin is pressure. Fluid pressure increases with depth, so the bottom of a submerged object sits in higher-pressure fluid than the top. The pressure pushing up on the underside therefore exceeds the pressure pushing down on the topside, and the net result is an upward force. Working through the pressure integral over the whole surface gives exactly rho times g times the displaced volume — the weight of the missing fluid.
To predict the object's fate, compare the buoyant force with the object's own weight, W = rho_object times g times V_object:
W < F_b -> the object rises and floats
W = F_b -> the object is neutrally buoyant and hovers
W > F_b -> the object sinks
For a fully submerged body, V_displaced equals the object's whole volume, so the comparison reduces to a contest between the fluid density and the object's average density. Anything denser than the fluid sinks; anything less dense floats. A floating object settles at the depth where the displaced volume produces just enough buoyancy to match its weight.
A worked example
Take a solid object with a volume of V = 0.01 m^3, fully submerged in fresh water. The density of water is rho = 1000 kg/m^3, and g = 9.81 m/s^2. Find the buoyant force, and decide whether the object floats.
Step 1 — identify the displaced volume. The object is fully submerged, so the volume of water it displaces equals its own entire volume:
V_displaced = V = 0.01 m^3
Step 2 — apply Archimedes' principle.
F_b = rho_fluid * g * V_displaced
F_b = 1000 * 9.81 * 0.01
F_b = 98.1 N
So the water pushes up on this object with a force of 98.1 N, regardless of what the object is made of. The buoyant force depends only on the fluid and the displaced volume.
Step 3 — compare with the object's weight. Now the material matters. If the object's own weight is less than 98.1 N — meaning its average density is below 1000 kg/m^3 — the upward buoyant force wins and the object rises until it floats partly out of the water. If its weight exceeds 98.1 N, gravity wins and it sinks. If the weight is exactly 98.1 N, the object hovers at any depth, neutrally buoyant. A weight of 98.1 N corresponds to a mass of 10 kg, so the cut-off here is whether the 0.01 m^3 object masses more or less than 10 kg.
Common mistakes
Using the object's volume when it is only partly submerged. For a floating body, V_displaced is the submerged volume, not the total volume. A log floating with half its bulk above the waterline displaces only the lower half. Plugging in the full volume overestimates the buoyant force.
Bringing the object's density into the buoyant force. F_b depends on the fluid density and the displaced volume — never on what the object is made of. The object's density enters only when you compute its weight for the float-or-sink comparison.
Forgetting that fluid density varies. Seawater is denser than fresh water, so a hull floats slightly higher in the sea than in a river. Warm water is less dense than cold. The same object can float in brine and sink in fresh water if its density sits between the two.
Ignoring trapped air. A sealed hollow object, or a hull with watertight compartments, displaces its full external volume while weighing very little. That is the whole reason a steel ship floats. If the compartments flood, the displaced volume effectively collapses and the buoyancy disappears.
Confusing buoyant force with net force. The buoyant force is only the upward part of the picture. Whether the object actually accelerates depends on F_b minus the weight. A submerged rock still feels 98.1 N of buoyancy per 0.01 m^3 — it just feels even more weight pulling the other way.
Try the interactive NovaSolver calculator
Running one case by hand is straightforward, but watching the float-or-sink threshold shift as you change density and fluid makes the principle stick. The Archimedes Buoyancy Simulator on NovaSolver lets you set the object density, its volume, the submersion fraction, and the fluid — fresh water, seawater, or mercury — and it returns the buoyant force, the weight, the net force, the buoyancy-to-weight ratio, and a clear float, sink, or neutral status.
Related calculators
- Buoyancy Calculator — a focused tool for the buoyant force alone, handy when you just need a quick number.
- Buoyancy & Stability Simulator — goes a step further to check whether a floating body is stable or will capsize.
- Hydrostatic Pressure Simulator — for the depth-dependent pressure that is the physical source of buoyancy in the first place.
You can browse the full set in the fluid mechanics tools hub.
Closing note
Archimedes' principle is one of the oldest results in physics that working engineers still reach for every day. The buoyant force is just the weight of the displaced fluid — rho times g times volume — and an object floats when that force can match its weight. Three ideas carry most of the practical work: use the displaced volume, not the total volume; keep the fluid density and the object density in their separate roles; and remember that trapped air is what lets heavy hulls float. Run the comparison, and the behaviour of almost any body in a fluid becomes predictable.
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