Hydrostatic pressure Equation, Derivation & Rearrangements

Core idea

Overview

Pressure in a still fluid increases with depth because the fluid above the point has weight. The total pressure equals the reference or surface pressure plus the hydrostatic contribution from the fluid column.

When to use: Use this for fluids at rest when density is nearly constant and depth is measured downward from a reference surface.

Why it matters: Hydrostatic pressure is the basis for tank design, depth gauges, dams, and many pressure measurements in process equipment.

Symbols

Variables

P = Pressure, $P_{0}$ = Reference Pressure, $ρ$ = Fluid Density, g = Gravitational Acceleration, h = Depth

P

Pressure

Pa

P_{0}

Reference Pressure

Pa

ρ

Fluid Density

k g / m^{3}

g

Gravitational Acceleration

m / s^{2}

h

Depth

m

Walkthrough

Derivation

Derivation of Hydrostatic pressure

The equation follows from balancing forces on a small static fluid column.

The fluid is at rest.
Density is constant over the depth considered.
Gravity is uniform.

1

Balance a fluid column

The pressure increase supports the weight of the fluid column above the point.

Δ P A = ρ A h g

2

Cancel area

The cross-sectional area cancels, so only density, gravity, and vertical depth matter.

Δ P = ρ g h

3

Add the reference pressure

Total pressure equals the reference pressure plus the hydrostatic increase.

P = P_{0} + ρ g h

Result

P = P_{0} + ρ g h

Source: OpenStax University Physics Volume 1, Pressure in Fluids, accessed 2026-04-09; Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013

Free formulas

Rearrangements

Solve for $P_{0} = P - ρ g h$

Reference Pressure

P_{0} = P - ρ g h

Subtract the hydrostatic pressure term from the total pressure to find the surface pressure.

Difficulty: 2/5

Solve for $h = \frac{P - P _{0}}{ρ g}$

Depth

\frac{P - P _{0}}{ρ g} = h

Isolate depth by subtracting surface pressure and dividing by the remaining factors.

Difficulty: 3/5

Solve for $ρ = \frac{P - P _{0}}{g h}$

Fluid Density

\frac{P - P _{0}}{g h} = ρ

Isolate density by subtracting surface pressure and dividing by the product of gravity and depth.

Difficulty: 3/5

Solve for $g = \frac{P - P _{0}}{ρ h}$

Gravitational Acceleration

\frac{P - P _{0}}{ρ h} = g

Isolate gravitational acceleration by subtracting surface pressure and dividing by the density-depth product.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph shows that hydrostatic pressure (P) increases linearly with depth (h), forming a straight line with a positive slope. For a student, this means that the deeper you go, the more the pressure builds up at a steady rate. The most important feature is this direct, linear relationship between depth and pressure. This illustrates how pressure in a fluid grows consistently as you descend.

Graph type: linear

Why it behaves this way

Intuition

Pressure grows linearly with depth because each extra metre adds another slab of fluid weight above the point.

P

Pressure at depth

The pressure you feel at the chosen point.

P_{0}

Reference pressure

The pressure already present at the surface or starting point.

ρ g h

Hydrostatic pressure increase

The weight per area of the fluid column.

Free study cues

Insight

Canonical usage

This equation is used to calculate pressure in a fluid at a specific depth, where all variables are expressed in consistent units within a chosen system.

Common confusion

Students often confuse units between the Imperial/US customary system and the SI system, leading to incorrect pressure calculations. For example, using density in kg/ $m^{3}$ with depth in feet without conversion.

Dimension note

This equation does not produce a dimensionless quantity; it calculates pressure, which has physical dimensions.

Unit systems

$P$ Pa · The resulting pressure unit is Pascals (Pa) when SI base units are used for all input variables.

$P_{0}$ Pa · The reference pressure at the surface, typically atmospheric pressure, should be in Pascals (Pa) for SI consistency.

$ρ$ kg/m^3 · Fluid density must be in kilograms per cubic meter (kg/m^3) for SI calculations.

$g$ m/s^2 · Gravitational acceleration should be in meters per second squared (m/s^2).

$h$ m · Depth must be in meters (m) for SI unit consistency.

Ballpark figures

Quantity:
Quantity:

One free problem

Practice Problem

Water has density 1000 kg/ $m^{3}$ . If surface pressure is 101325 Pa, g is 9.81 m/ $s^{2}$ , and depth is 3.0 m, what is the pressure?

Reference Pressure101325 Pa

Fluid Density1000 kg/m^3

Gravitational Acceleration9.81 m/s^2

Depth3 m

Solve for: pressure

Hint: Add rho g h to the surface pressure.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating the pressure at the bottom connection of a liquid storage tank, Hydrostatic pressure is used to calculate Pressure from Reference Pressure, Fluid Density, and Gravitational Acceleration. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Study smarter

Tips

Use absolute pressure if the downstream equation needs absolute pressure.
Use gauge pressure consistently if the reference pressure is taken as zero.
Depth is vertical distance, not distance along a sloped wall.

Avoid these traps

Common Mistakes

Forgetting to include the reference pressure.
Using horizontal distance instead of vertical depth.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The equation follows from balancing forces on a small static fluid column.

Use this for fluids at rest when density is nearly constant and depth is measured downward from a reference surface.

Hydrostatic pressure is the basis for tank design, depth gauges, dams, and many pressure measurements in process equipment.

Forgetting to include the reference pressure. Using horizontal distance instead of vertical depth.

When estimating the pressure at the bottom connection of a liquid storage tank, Hydrostatic pressure is used to calculate Pressure from Reference Pressure, Fluid Density, and Gravitational Acceleration. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Use absolute pressure if the downstream equation needs absolute pressure. Use gauge pressure consistently if the reference pressure is taken as zero. Depth is vertical distance, not distance along a sloped wall.

References

Sources

OpenStax University Physics Volume 1, Pressure in Fluids, accessed 2026-04-09
OpenStax University Physics Volume 1, Measuring Pressure, accessed 2026-04-09
Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013
NIST CODATA
IUPAC Gold Book
Wikipedia: Hydrostatic pressure
Britannica: Fluid statics
IUPAC Gold Book: Continuum hypothesis

Hydrostatic pressure

Overview

Variables

Derivation

Balance a fluid column

Cancel area

Add the reference pressure

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Hydrostatic head

General manometer

Differential manometer

Manometer for gas

Frequently Asked Questions

Sources