Buoyancy Calculator

What buoyancy is, and where the force comes from

Push a sealed container under water and the water pushes back. Pressure grows with depth, so the water pressing on the bottom of the container is at a higher pressure than the water pressing on the top. Adding the pressure over the whole surface leaves a net upward force, and that net force is buoyancy.

Archimedes pinned down its size more than 2,000 years ago: the buoyant force on a submerged object equals the weight of the fluid that object displaces. The NASA Glenn buoyancy lesson makes the point with an aluminum cube. The surrounding water cannot tell the cube from the water it replaced, so it supports the cube exactly as it would support that same volume of water, which is the weight of the displaced water.

Two consequences follow. The force depends on the displaced volume and the fluid, not on what the object is made of, and it does not change with depth, because water is close to incompressible and its density barely moves between the surface and the bottom of a pool or a lake. A rock at 1 m and the same rock at 10 m feel the same lift.

The formula and its units

F_b = fluid density x displaced volume x g, with density in kilograms per cubic meter, volume in cubic meters and g = 9.81 m/s2. The answer comes out in newtons, and dividing by 4.44822 converts it to pounds of force.

The volume that matters is the volume under the waterline, not the whole object. A floating hull displaces only the part that is wet, which is why a boat can weigh 400 kg and still sit in shallow water. It is also why the object's own density never appears in the formula: the fluid does the pushing, and the object's mass only decides how deep it settles.

Fresh water is close to 1,000 kg/m3, and the USGS Water Science School puts the maximum near 4 degrees C at about 1,000 kg/m3, with warmer water slightly lighter. Sea water is denser: the NOAA JetStream page on sea water explains how dissolved salt raises density, and a working figure is 1,025 kg/m3, about 2.5 percent more than fresh. That gap is why ships ride a little higher in the ocean than in a river.

Example: a 10 L bucket in a lake

Take a 10 liter bucket, which is 0.01 cubic meters. Pushed fully under fresh water it displaces 10 L, so the buoyant force is 1,000 x 0.01 x 9.81 = 98.1 N, or 98.1 / 4.44822 = 22.05 lbf. That lift is the same whether the bucket holds air, water or concrete, because the fluid only sees the outside shape.

What changes is the bucket's own weight. An empty 5 kg bucket weighs 5 x 9.81 = 49.05 N. The 98.1 N of lift available is more than that, so the bucket floats, and it sinks only until the displaced water weighs 49.05 N, which happens at 5 L submerged: exactly half the bucket. In sea water at 1,025 kg/m3 the same 5 kg needs 5 / 1,025 = 0.004878 m3 of water under the surface, 48.78 percent of the bucket, so it floats slightly higher.

Fill the bucket with lake water and it displaces 10 L of itself: weight 98.1 N, buoyancy 98.1 N, a perfect match. Push it under by hand and it stays where you leave it, which is what divers mean by neutral buoyancy and what they tune with a weight belt.

Float, sink, or hover: the density test

You do not need the force to predict the outcome, only the two densities. An object floats when its average density is lower than the fluid's, sinks when it is higher, and hovers when the two match. When it floats, the submerged fraction equals the object density divided by the fluid density, so the closer the two numbers are, the deeper the object sits.

Ice is the case everyone has seen. At about 917 kg/m3 an ice cube floats with 917 / 1,000 = 0.917, or 91.7 percent of its volume below the surface, which leaves less than a tenth showing. An iceberg obeys the same ratio, and the small visible part is why they are so easy to misjudge from a boat.

Steel at 7,850 kg/m3 sinks as a solid block, yet a ship built from the same steel floats. The hull encloses air, so the average density of the whole floating volume, steel plus air, lands below 1,000. Every hollow hull relies on that averaging, and it is why a holed ship floods and then sinks: the air leaves, the average density climbs, and the waterline rises until the deck goes under.

Apparent weight underwater

Subtract the buoyant force from the weight in air and you have the apparent weight, which is what a scale reads underwater: apparent weight = weight - buoyant force. That number matters when you lift an anchor by hand, size a diver's weight belt, or check what a crane is holding as it lowers a load into water.

A granite block of 10 L has a mass of about 27 kg at 2,700 kg/m3. In air it weighs 27 x 9.81 = 264.87 N. Under water it still displaces 10 L, so buoyancy removes 98.1 N and 264.87 - 98.1 = 166.77 N remains, which reads as 166.77 / 9.81 = 17 kg on a scale marked in kilograms. The block has not changed. Ten kilograms of water are carrying part of it.

Watch the direction of the subtraction. If an object floats, the lift at full submersion is larger than its weight, so apparent weight comes out negative, and that negative number is the extra force you must supply to hold it down. It is why a beach ball is awkward to keep under the surface and why a life jacket pushes a swimmer back up.

The same numbers in pounds and gallons

US units carry the same physics with different constants. One cubic foot of fresh water weighs 62.4 lb, and one US gallon weighs 8.34 lb. A 55 gallon drum, the standard dock float, displaces 55 x 8.34 = 458.7 lb of fresh water when pushed fully under, and 55 x 8.55 = 470.25 lb in sea water, since salt water weighs about 8.55 lb per gallon.

That displaced weight is the float capacity, and it is never the working load. A drum half submerged still has reserve buoyancy for waves, spray and an unevenly loaded deck, so flotation for docks and small craft is normally sized to carry the load at about half submersion. For the drum above, treat roughly 230 lb as the load and keep the rest in reserve.

Conversions for the calculator: 1 US gallon is 3.785 L, 1 cubic foot is 28.317 L, and 1 lb is 0.4536 kg. Enter the mass and volume in whichever units you have and the tool converts them before applying the formula, so a 55 gallon drum can go in as 55 gal without a separate step.

Boats, load limits and the waterline

Everything a boat carries ends up as displaced water. A jon boat with a 120 kg hull carrying 300 kg of gear and people has to displace 420 kg of water to float, which is 420 L of fresh water below the waterline. Whether that is safe depends on the hull volume: a hull of 1,200 L total leaves 780 L of reserve above the waterline, and the visible band of hull between water and deck, the freeboard, is the margin you can see.

Load lines on commercial ships turn the same idea into regulation. More cargo means deeper displacement, less freeboard and less reserve, so hulls carry marks for fresh water, sea water, summer and winter, because the 2.5 percent density gap between fresh and salt water shifts the waterline on its own with no cargo change at all.

For everyday jobs the calculator above answers the practical questions directly: will it float, how deep will it sit, and what will it weigh once it is under. For the geometry side, the volume calculator works out the space inside a hull or tank, and the aquarium tank weight calculator applies the same water weight constants to a full tank.