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Archimedes Principle Calculator

Calculate the buoyant force on an object submerged in fluid

Archimedes Principle

Buoyant Force: Fb = rho * g * V

Where:

  • rho = fluid density (kg/m³)
  • g = gravitational acceleration (m/s²)
  • V = volume of displaced fluid (m³)

An object floats if the buoyant force exceeds its weight.

What Is Archimedes' Principle?

Archimedes' principle states that any object partially or fully submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. Discovered by the Greek mathematician Archimedes of Syracuse around 250 BCE, this principle explains why ships float, helium balloons rise, and submarines can control their depth.

The buoyant force Fb = ρ × g × V depends on only three factors: the fluid density ρ (kg/m³), gravitational acceleration g (m/s²), and the volume of displaced fluid V (m³). Importantly, the buoyant force does not depend on the object's own density — only on the fluid and the submerged volume. This is why a steel ship can float: the hollow hull displaces a large volume of water whose weight exceeds the ship's own weight.

The Buoyant Force Formula

The buoyant force is straightforward to calculate once you know the displaced volume and fluid density. For a fully submerged object, V equals the object's total volume. For a floating object, V equals only the submerged portion.

Archimedes' Principle

Fb = ρ × g × V Apparent weight = W - Fb Object floats when Fb ≥ W

Where:

  • Fb= Buoyant force — upward force exerted by the fluid (Newtons, N)
  • ρ= Fluid density (kg/m³). Water = 1000, seawater = 1025, air = 1.225
  • g= Gravitational acceleration (m/s²). Earth = 9.81, Moon = 1.62
  • V= Volume of displaced fluid (m³) — equals submerged volume of object
  • W= Weight of the object in air (N). W = m × g

Does It Float or Sink?

The key question Archimedes' principle answers: will an object float? The answer depends on comparing the buoyant force Fb to the object's weight W:

ConditionResultEquivalent Density Rule
Fb > WObject rises and floatsρ_object < ρ_fluid
Fb = WNeutral buoyancy (hovers)ρ_object = ρ_fluid
Fb < WObject sinksρ_object > ρ_fluid

How to Use This Calculator

Calculate buoyant force and determine if an object will float:

  1. Enter Fluid Density: Input the density in kg/m³. Defaults to 1000 for fresh water. Use 1025 for seawater, 1.225 for air (at sea level), 1260 for glycerin, or 13,600 for mercury.
  2. Enter Displaced Volume: Input the volume of fluid displaced in m³. For a fully submerged object, this equals the object's total volume. For a floating object, this equals only the submerged volume.
  3. Set Gravity: Default is 9.81 m/s² (Earth). Change to 1.62 for Moon calculations or 3.71 for Mars.
  4. Optional — Object Weight: Enter the object's actual weight in Newtons (N) to get the apparent weight when submerged and a direct float/sink determination with a color-coded result.

Real-World Applications

Archimedes' principle is fundamental to naval architecture and ship design. Naval architects calculate hull displacement to ensure a vessel floats at the correct waterline. Cargo ships use the Plimsoll line — markings on the hull showing maximum safe loading depth based on water density (fresh water at 1000 kg/m³ versus seawater at 1025 kg/m³). A ship loaded for fresh water that enters the ocean will rise slightly because salt water is denser, changing the displaced volume needed to support the same weight.

In aviation and atmospheric science, buoyancy governs hot air balloons (heated air is less dense than surrounding air), weather balloons that carry instruments to the stratosphere, and the behavior of smoke plumes from wildfires and volcanic eruptions. Even the largest objects in the universe — interstellar gas clouds — follow Archimedes' principle in determining whether they rise or sink within a galaxy.

Worked Examples

Submerged Ball in Water

Problem:

A ball of volume 0.005 m³ (5 liters) is fully submerged in fresh water (ρ = 1000 kg/m³). What is the buoyant force, and will it float if it weighs 30 N?

Solution Steps:

  1. 1Buoyant force: Fb = ρ × g × V = 1000 × 9.81 × 0.005 = 49.05 N
  2. 2Mass of displaced water: 1000 × 0.005 = 5 kg
  3. 3Compare: Fb (49.05 N) > W (30 N)
  4. 4Apparent weight: 30 - 49.05 = -19.05 N (negative means it rises)
  5. 5Since Fb > W, the ball will float — it will rise until the displaced volume drops enough for Fb = W
  6. 6At equilibrium: 1000 × 9.81 × V = 30, so V = 30/9810 = 0.00306 m³ submerged

Result:

Buoyant force = 49.05 N. The ball floats with about 61% of its volume submerged (0.00306 m³ / 0.005 m³).

Iceberg in Seawater

Problem:

An iceberg has an average density of 917 kg/m³ floating in seawater (ρ = 1025 kg/m³). What fraction of its volume is submerged?

Solution Steps:

  1. 1At equilibrium: Fb = W, so ρ_seawater × g × V_submerged = ρ_ice × g × V_total
  2. 2Cancel g and solve: V_submerged / V_total = ρ_ice / ρ_seawater = 917 / 1025
  3. 3Compute: 917/1025 = 0.895 = 89.5%
  4. 4This is why icebergs show only about 10% above water — 'tip of the iceberg' is scientifically accurate

Result:

89.5% of the iceberg's volume is below the waterline. Only about 10.5% is visible above the surface, illustrating why icebergs are so dangerous to ships.

Hot Air Balloon

Problem:

A hot air balloon has a volume of 2,800 m³. The outside air density is 1.225 kg/m³ at 15°C, and the heated air inside has density 0.95 kg/m³. What is the net buoyant force (lifting force)?

Solution Steps:

  1. 1Buoyant force equals weight of displaced outside air: Fb = 1.225 × 9.81 × 2800 = 33,665 N
  2. 2Weight of hot air inside: W_inside = 0.95 × 9.81 × 2800 = 26,106 N
  3. 3Net lifting force = Fb - W_inside = 33,665 - 26,106 = 7,559 N
  4. 4In terms of mass: 7,559 / 9.81 ≈ 771 kg of payload (balloon + basket + passengers)

Result:

Net lifting force = 7,559 N, enough to lift approximately 771 kg total (balloon envelope, basket, burner, fuel, and passengers).

Tips & Best Practices

  • ✓Fresh water density = 1000 kg/m³, seawater = 1025 kg/m³ — the 2.5% difference matters for precision marine calculations
  • ✓The buoyant force depends only on displaced fluid, not on the object's material — a 1 m³ block of steel and a 1 m³ block of cork experience the same Fb when fully submerged
  • ✓Apparent weight can be negative — this means the object rises, not because it has anti-gravity, but because Fb > W
  • ✓For irregular objects, measure displaced volume by water displacement — submerge the object and measure the water level rise
  • ✓1 m³ of water has a mass of 1,000 kg and a weight of 9,810 N on Earth — useful for quick estimates

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Frequently Asked Questions

A solid steel block sinks because its density (~7,800 kg/m³) far exceeds water's density (1,000 kg/m³). But a ship is not solid — it's a hollow shell that displaces a huge volume of water. The ship's average density (total mass ÷ total enclosed volume, which is mostly air) must be less than water's density. A cargo ship with a mass of 100,000 tonnes needs to displace at least 100,000 m³ of water to float — which is why ship hulls are so large.
Buoyant force is the specific upward force (in Newtons) exerted by the fluid on the object. Buoyancy is the general phenomenon or principle. The buoyant force equals ρgV — it's a concrete number you can calculate. When people say something 'has good buoyancy,' they typically mean it has low density relative to the fluid, so the buoyant force easily exceeds its weight.
Submarines use ballast tanks to change their average density. To dive, they flood tanks with seawater, increasing total mass while keeping volume constant — this makes average density exceed water density. To surface, they pump compressed air into the tanks, expelling water and reducing mass until average density drops below water density. To hover at a specific depth (neutral buoyancy), they adjust until average density exactly equals seawater's density. This is Archimedes' principle in action.
In microgravity (free fall), the buoyant force becomes zero because g = 0 — the formula Fb = ρgV goes to zero. Fluids don't settle and there's no 'up' for buoyancy to act toward. This has practical consequences: boiling looks completely different in space (no bubbles rise), fuel tanks need special mechanisms (capillary action or bladders) to separate liquid from gas, and astronauts can't rely on buoyancy to determine orientation in water.
Apparent weight is what a scale would read if you weighed an object while it was submerged in a fluid. It equals the object's true weight minus the buoyant force: W_apparent = W - Fb. If Fb > W, apparent weight goes negative — the object rises. Divers feel lighter underwater because their bodies displace water, producing a buoyant force that partially counteracts gravity. The same principle explains why recovering objects from the ocean floor is easier than their weight suggests.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: University Physics

by Young & Freedman

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.