Torricelli's Theorem Calculator

Open a hole in the side of a water tank and the water shoots out with a speed that depends only on how deep below the surface the hole is. Not on the size of the hole. Not on the volume of water in the tank. Not on what’s in the water (within reason). Just the depth. Evangelista Torricelli figured this out in 1643, and it remains one of the most elegant results in classical fluid mechanics.

The formula:

v = √(2 · g · h)

Where v is the exit velocity at the opening (m/s), g is gravitational acceleration (9.81 m/s² on Earth), and h is the vertical distance from the fluid surface down to the opening (m).

Why this works:

Torricelli’s theorem is what you get when you apply Bernoulli’s equation along a streamline from the fluid surface to the opening. At the surface, the fluid is essentially still and the pressure is atmospheric. At the opening, the fluid is moving at speed v and the pressure is again atmospheric (it has just exited into the atmosphere). The difference in elevation is h. Bernoulli says ½v² = g·h, which gives v = √(2gh).

That formula is identical to the speed of a freely falling object dropped from height h. This is not a coincidence: the fluid is essentially in free-fall once you account for the pressure gradient. Energy conservation handles the rest.

Three assumptions baked in:

1. The tank is wide compared to the opening, so the surface falls slowly. If the tank is narrow, the surface drops fast and the velocity at the opening becomes time-dependent.
2. The fluid is inviscid (no viscosity). Real water has some viscosity, but for the typical kitchen-sink to small-reservoir scale, it’s negligible.
3. The opening is small and exposed to atmospheric pressure. If you connect a hose to the opening that’s not at atmospheric pressure, the formula needs modification.

Discharge coefficient — what real openings actually do:

The ideal v = √(2gh) is the maximum possible velocity. Real fluids exiting through a real opening achieve less than that, by a factor C_d called the discharge coefficient. The actual flow rate Q is:

Q = C_d · A · √(2 · g · h)

Where A is the cross-sectional area of the opening and C_d depends on geometry:

The reason for C_d < 1: when fluid leaves a sharp-edged opening, the streamlines converge inside the hole before reaching equilibrium. This contracted region (the vena contracta) has area smaller than A, so the effective area is reduced. Rounding the opening straightens the streamlines and brings C_d up toward 1.

Worked example, water tank with a small hole:

A water tank has the surface 3 m above a small sharp-edged hole. What is the ideal exit velocity?

v = √(2 · 9.81 · 3) = √58.86 = 7.67 m/s

For a sharp-edged hole 10 mm in diameter (A = 78.5 mm² = 7.85 × 10⁻⁵ m²), the actual flow rate with C_d = 0.61:

Q = 0.61 × 7.85 × 10⁻⁵ × 7.67 = 3.67 × 10⁻⁴ m³/s ≈ 22 L/min

Tank-draining time (the slow-tank-empty problem):

For a tank of constant cross-section A_t draining through an opening A through a depth h, conservation of volume gives:

A_t · dh/dt = −C_d · A · √(2gh)

Integrating from h₀ (initial) to 0 (empty):

t = (A_t / (C_d · A)) · √(2 · h₀ / g)

For our 3 m example, if the tank is 1 m² cross-section and the hole is 78.5 mm²:

t = (1 / (0.61 · 7.85 × 10⁻⁵)) · √(2 · 3 / 9.81) = 20,880 · 0.782 = 16,330 s ≈ 4.5 hours

That’s why slow leaks take so long — the velocity is fastest at the start and slows as the tank empties.

Where the theorem holds up well:

• Aquariums, kitchen sinks, swimming pools, bathtubs.
• Garden hoses with an elevated water source (gravity-fed water from a tank uphill).
• Designing emergency drains for storage tanks, fire-suppression systems.
• Estimating flow rate from a punctured container in a leak scenario.

Where it breaks down:

• Pressurized vessels: Torricelli only applies to fluids open to atmospheric pressure at the surface. A pressurized tank delivers higher v, with the added pressure converting to extra kinetic energy by Bernoulli’s full equation: v = √(2(p_gauge + ρgh)/ρ) for incompressible fluids.
• Compressible gases: For air leaving a pressure vessel, you need compressible flow theory; at high pressure ratios the flow chokes at the speed of sound and Torricelli’s formula no longer applies.
• Viscous fluids: Honey, oil, syrup. The viscosity term in the full Navier-Stokes equations dominates; effective C_d is much smaller, and depending on Re may go below 0.1.
• Surface tension dominates at small scales: For openings under about 1 mm in water, surface tension significantly alters the flow.

Connection to other phenomena:

Torricelli’s theorem is essentially the fluid version of the kinematics formula v² = 2g·h that applies to a falling rock. Both come from energy conservation. The classroom demonstration where a side of a tall water column shoots a stream out at the bottom matches the predicted parabolic trajectory of a horizontally-launched projectile with the calculated speed — a satisfying confirmation that the underlying physics is one and the same.