Pressure Variation with Depth — the Hydrostatic Equation

Why the deep end of a pool pushes harder on your ears than the shallow end — and exactly how much harder.

Fluid StaticsMechanical Engineering Year 1Free preview
⏱️ About 16 min

Dive to the bottom of a 3 m pool and your ears feel it immediately — but a 3 m-wide pool and a 3 m-wide bathtub push on your eardrums with exactly the same pressure at the same depth (and therefore the same force, since your eardrum's area is the same either way). Shape doesn't matter. Depth does. That single fact is the entire hydrostatic pressure equation.

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The big idea: In a fluid at rest, pressure increases linearly with depth because each layer of fluid has to support the weight of everything above it. The relationship is p = p0 + ρgh — and it depends only on depth h, fluid density ρ, and gravity g, never on the shape or width of the container.
🎯 By the end, you'll be able to
  • Derive the hydrostatic equation p = p0 + ρgh from a force balance on a fluid column
  • Explain why pressure at a given depth is independent of the container's shape ('the hydrostatic paradox')
  • Distinguish absolute pressure from gauge pressure in the equation
  • Calculate pressure at any depth in a fluid, given density and reference pressure
📎 Helpful to know first
  • Pressure — Definition, Units, Absolute vs Gauge

A force balance on a column of fluid

Imagine a thin vertical column of fluid inside a larger tank, extending from the free surface down to some depth h. That column isn't accelerating — it's just sitting there — so the forces on it must balance. Three things push on the column: atmospheric pressure pushing down on its top, the fluid below pushing up on its bottom, and its own weight pulling it down.

Balancing those forces and dividing through by the column's cross-sectional area (which cancels out completely — that's the key step) leaves a relationship between pressure and depth alone.

\[ p = p_0 + \rho g h \]
p0 = pressure at the reference point (often atmospheric, at the free surface). ρ = fluid density. g = gravitational acceleration. h = depth below the reference point.
A tank of fluid showing pressure increasing linearly with depth, marked at two points h1 and h2 below the free surface free surface (p = p_atm) 1 h1 2 h2 depth Point 2 is deeper, so p2 > p1 — pressure grows linearly with depth.
Two points in the same connected fluid: the deeper point (2) always has higher pressure than the shallower point (1), and the difference depends only on the vertical distance between them.
🔑 Shape doesn't matter — only depth and density do

The derivation used an imagined vertical fluid column of area A, and A canceled out completely. That means pressure at a given depth doesn't depend on the size or shape of the container at all. A narrow test tube and a wide swimming pool produce identical pressure at the same depth, as long as the fluid and the depth are the same.

This is closely related to the classic hydrostatic paradox: even though pressure at a given depth is shape-independent, the total force on a container's base (force = pressure × area) can be much larger or smaller than the actual weight of fluid it holds, depending on the container's shape — a result that startled 17th-century scientists.

⚠️ Absolute vs. gauge — check which one your problem wants

If p0 is absolute atmospheric pressure (about 101.3 kPa at sea level), then p from the equation is absolute pressure. If you set p0 = 0 (treating the free surface as your zero reference, which is how most pressure gauges are calibrated), then p is gauge pressure — the number a pressure gauge would actually read. Most engineering problems use gauge pressure; always check which one is being asked for.

🎮 Interactive: pressure vs. depth in a fluid column LIVE
Predict first: If you double the depth, does the gauge pressure double too? Test it before reading on.

An interactive simulator showing a vertical fluid column where the user can drag a depth marker and see gauge pressure computed live from p = rho * g * h, with a selectable fluid density.

Drag the depth marker down the column and watch gauge pressure update live. Switch fluids to see how density changes the slope of the line.
📝 Worked example: A tank is filled with water (ρ = 1000 kg/m³). Find the gauge pressure at a depth of 4 m below the free surface. Use g = 9.81 m/s².
  1. Use the hydrostatic equation with gauge pressure, so p0 = 0 at the free surface: p = ρgh.
  2. Substitute values: p = (1000 kg/m³)(9.81 m/s²)(4 m).
  3. Compute: p = 39{,}240 Pa = 39.24 kPa.
✓ p ≈ 39.2 kPa (gauge)
✏️ Practice: Find the gauge pressure, in kPa, at a depth of 2.5 m in seawater (ρ = 1025 kg/m³, g = 9.81 m/s²).
kPa
Solution
  1. p = ρgh = (1025)(9.81)(2.5).
  2. p = 25{,}138 Pa ≈ 25.1 kPa.
✏️ Practice: A pressure gauge at the bottom of a water tank (ρ = 1000 kg/m³) reads 58.86 kPa. How deep is the tank, in metres? Use g = 9.81 m/s².
m
Solution
  1. Rearrange p = ρgh for h: h = p / (ρg).
  2. h = 58{,}860 / (1000 × 9.81) = 6.0 m.

Check your understanding

1. A narrow test tube and a wide swimming pool are both filled with water to the same depth. How do the pressures at the bottom compare?
Pressure depends only on depth and density, never on the container's shape or width — this is the hydrostatic paradox.
2. If you double the depth in the same fluid, what happens to the gauge pressure?
p = ρgh is a linear relationship in h, so doubling depth exactly doubles the gauge pressure.
3. A pressure gauge reads 'gauge pressure' relative to what reference?
Gauge pressure is measured relative to local atmospheric pressure — that's why gauge pressure at the free surface of an open tank reads zero.
🔎 Related lessons
Pressure — Definition, Units, Absolute vs Gauge · soon Manometry & Pressure Measurement Buoyancy & Archimedes' Principle · soon
✅ Key takeaways
  • The hydrostatic equation p = p0 + ρgh gives pressure at any depth in a fluid at rest.
  • Pressure depends only on depth and fluid density — never on the container's shape (the hydrostatic paradox).
  • Gauge pressure sets p0 = 0 at the free surface; absolute pressure includes atmospheric pressure.
  • Pressure grows linearly with depth: double the depth, double the gauge pressure.
➡️ Now that you can find pressure at a point, the next lesson uses exactly this equation to read a manometer — one of the simplest and most common pressure-measuring instruments in engineering.
Want to test yourself on this? Try the Mechanical Aptitude test →
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