Buoyancy calculator: upthrust and whether it floats

The buoyant force (called upthrust in the UK) on an object in a fluid equals the weight of the fluid it pushes out of the way. That is Archimedes' principle, and as a formula it is F_b = ρVg: the fluid's density times the volume displaced times g. A 0.02 m³ block under water feels 1,000 × 0.02 × 9.81 = 196.2 N of buoyant force. This buoyancy calculator is part of our Pressure Calculator: it finds the buoyant force, says whether the object would rise or sink, and works backwards to the volume displaced or the fluid's density.

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Pressure Calculator

Calculate pressure in solids and fluids, including upthrust

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How to use the buoyancy calculator

1. Open the Pressure Calculator and choose Upthrust on the top row of Calculate. Then choose what to find: Upthrust, Volume (of fluid displaced) or Density (of the fluid).
2. Type the fluid density, or pick it from Common values: air, petrol (gasoline), olive oil, water, seawater or mercury.
3. Type the volume of fluid displaced. For an object fully under the surface, that is the object's whole volume. Volume can be in mm³, cm³, mL, L or m³.
4. Leave Gravity blank for 9.81 m/s², or pick a planet.
5. To see whether it floats, type the object's weight in newtons in Weight of the object. It is optional.
6. Read the buoyant force with its working, the mass of fluid displaced and, with a weight, the object's weight in the fluid and whether, fully submerged, it would rise or sink. The diagram draws the two forces to scale.

The buoyant force formula

The calculator writes it as U = ρVg, with U for upthrust. ρV is the mass of fluid displaced, so ρVg is its weight. Rearranged, V = F_b ÷ (ρg) and ρ = F_b ÷ (Vg).

The force comes from pressure. Pressure in a fluid rises with depth (P = ρgh), so the fluid pushes harder on the bottom of an object than on its top. The difference is the buoyant force, and it always points up.

Float or sink?

• If the buoyant force when fully submerged is bigger than the weight, the object rises and ends up floating.
• If it is smaller, the object sinks. Its weight in the fluid, its apparent weight, is the weight minus the buoyant force.
• A floating object sinks until it displaces its own weight of fluid. Then the buoyant force equals the weight exactly.

The same test in terms of density: an object floats if its average density is less than the fluid's. The fraction of a floating object below the surface is its density divided by the fluid's.

Worked examples you can check in the calculator

Example 1: a block that sinks

A 0.02 m³ block weighing 500 N is lowered into water. Choose Upthrust, pick Water, type V = 0.02 (or 20 with the unit set to L) and a weight of 500.

The calculator shows 20 kg of water displaced, a weight in the water of 500 − 196.2 = 303.8 N, and that fully submerged it would sink. That 303.8 N is what a spring balance would read with the block hanging under water.

Example 2: a block that floats, and how much is under water

A 0.05 m³ block weighs 300 N. Fully under water, the buoyant force would be 1000 × 0.05 × 9.81 = 490.5 N, more than its weight, so the calculator says it would rise. It floats.

How much of it sits under the surface? Floating, the buoyant force equals the weight, 300 N. Choose Volume, type 300 and pick Water: V = 300 ÷ (1000 × 9.81) = 0.0306 m³. That is 0.0306 ÷ 0.05 = 61% of the block, as in the diagram. Its density is 300 ÷ 9.81 ÷ 0.05 = 612 kg/m³, and 612 ÷ 1000 gives the same 61%.

Example 3: how much water a boat displaces

A boat of mass 2,000 kg floats in the sea. Its weight is 2,000 × 9.81 = 19,620 N, so the sea must push up with 19,620 N. Choose Volume, type 19620 and pick Seawater: V = 19620 ÷ (1025 × 9.81) = 1.95 m³, which the calculator shows is 2,000 kg of seawater. In fresh water it would need 2.00 m³, so the boat floats a little lower in a river than at sea.

Example 4: the density of a liquid

A 500 cm³ metal block weighs 12.0 N in air and 8.0 N hanging in a liquid. The buoyant force is the difference, 12.0 − 8.0 = 4.0 N. Choose Density under Upthrust, type 4, set the volume unit to cm³ and type 500. The calculator gives ρ = 4 ÷ (0.0005 × 9.81) = 815 kg/m³: the liquid is less dense than water.

Example 5: buoyancy in air

Air is a fluid too. A hot air balloon's envelope holds 2,800 m³. Pick Air (1.2 kg/m³): F_b = 1.2 × 2800 × 9.81 = 32,962 N, the weight of 3,360 kg of air. The balloon rises if the envelope, basket, people and the hot air inside weigh less than that. Heating the air makes it less dense, which is what keeps the total under.

What the calculator doesn't do

• The floating depth directly. It tells you whether an object rises or sinks when fully submerged. For how much floats under, use the Volume mode with the weight as the buoyant force, as in Example 2.
• Volume from a shape. Work out the object's volume first, such as length × width × height for a block.
• Mass instead of weight. The weight field is in newtons. Multiply a mass in kg by 9.81 first.
• Partly submerged objects. V is only the part below the surface. Type that volume, not the whole object's.

Common mistakes

• Using the object's density. ρ in the formula is the fluid's density, not the object's.
• Using the whole volume of a floating object. Only the volume under the surface displaces fluid.
• Forgetting g. ρV is a mass in kg. Multiply by g for a force in newtons.
• cm³ in a formula that needs m³. 500 cm³ is 0.0005 m³. Set the volume unit, or convert first.
• Thinking heavy objects can't float. A steel ship floats because its hull displaces a large volume of water. What matters is average density.

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