Hydrostatic head Equation, Derivation & Rearrangements

Core idea

Overview

Hydrostatic head is the height of a fluid column that represents a pressure. For a constant-density fluid, pressure difference is density times gravity times head.

When to use: Use this when a pressure is reported as metres of fluid head or when converting a level difference to pressure.

Why it matters: Head units are common in pumps, tanks, and hydraulic systems because they describe pressure in terms of an equivalent fluid height.

Symbols

Variables

P = Pressure Difference, $ρ$ = Fluid Density, g = Gravitational Acceleration, $h_{p}$ = Pressure Head

P

Pressure Difference

Pa

ρ

Fluid Density

k g / m^{3}

g

Gravitational Acceleration

m / s^{2}

h_{p}

Pressure Head

m

Walkthrough

Derivation

Derivation of Hydrostatic head

Pressure head is the fluid-column height that would create the same pressure difference.

The fluid is static.
Density and gravity are constant.

1

Start with hydrostatic pressure

A static fluid column creates pressure equal to density times gravity times height.

Δ P = ρ g h

2

Name the height as pressure head

The head $h_{p}$ is the equivalent vertical height of the fluid column.

P = ρ g h_{p}

Result

P = ρ g h_{p}

Source: Engineering LibreTexts, 7.9: Fluid Statics, accessed 2026-04-09; OpenStax University Physics Volume 1, Pressure in Fluids, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $ρ = \frac{P}{g h _{p}}$

Fluid Density

ρ = \frac{P}{g h _{p}}

Rearrange the hydrostatic head equation to solve for density by dividing both sides by gravity and pressure head.

Difficulty: 2/5

Solve for $g = \frac{P}{ρ h _{p}}$

Gravitational Acceleration

g = \frac{P}{ρ h _{p}}

Rearrange the hydrostatic head equation to solve for gravitational acceleration by dividing both sides by density and pressure head.

Difficulty: 2/5

Solve for $h_{p} = \frac{P}{ρ g}$

Pressure Head

h_{p} = \frac{P}{ρ g}

Rearrange the hydrostatic head equation to solve for the pressure head by dividing both sides by the product of density and gravity.

Difficulty: 2/5

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

Visual intuition

Graph

The graph of pressure difference (P) versus fluid density (\rho) is a straight line passing through the origin, showing a direct proportional relationship. For a student, this means that if you double the density of a fluid, the pressure difference will also double, provided gravity and the pressure head remain the same. The most important feature is this linear increase, highlighting how fluid density directly impacts pressure. As density increases, pressure difference increases proportionally.

Graph type: linear

Why it behaves this way

Intuition

Hydrostatic head turns pressure into a height: a taller column of the same fluid means a larger pressure.

P

Pressure difference

The pressure represented by the fluid column.

h_{p}

Pressure head

The equivalent fluid height.

ρ g

Specific weight

How much pressure each metre of that fluid adds.

Free study cues

Insight

Canonical usage

This equation is used to calculate the pressure difference in a fluid column based on its density, gravitational acceleration, and height.

Common confusion

Students may confuse pressure head (a height) with pressure itself, leading to incorrect unit assignments.

Dimension note

This equation involves quantities with physical dimensions, so the result is not dimensionless.

Unit systems

$P$ Pa · Represents the pressure difference.

$ρ$ kg/m^3 · Represents the fluid density.

$g$ m/s^2 · Represents the local gravitational acceleration.

$h_{p}$ m · Represents the pressure head or height of the fluid column.

Ballpark figures

Quantity:
Quantity:
Quantity:

One free problem

Practice Problem

What pressure is equivalent to 12 m of water head if density is 1000 kg/ $m^{3}$ and g is 9.81 m/ $s^{2}$ ?

Fluid Density1000 kg/m^3

Gravitational Acceleration9.81 m/s^2

Pressure Head12 m

Solve for: pressure

Hint: Use P = rho g h.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When converting 12 m of water head into the pressure delivered at a pump suction line, Hydrostatic head is used to calculate Pressure Difference from Fluid Density, Gravitational Acceleration, and Pressure Head. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Study smarter

Tips

The head must be a vertical height.
Use the density of the fluid that provides the head.
Pressure head and pressure are proportional only when density and gravity are fixed.

Avoid these traps

Common Mistakes

Using the process fluid density when the head is actually a manometer-fluid height.
Forgetting that head in metres is not pressure until multiplied by rho g.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Pressure head is the fluid-column height that would create the same pressure difference.

Use this when a pressure is reported as metres of fluid head or when converting a level difference to pressure.

Head units are common in pumps, tanks, and hydraulic systems because they describe pressure in terms of an equivalent fluid height.

Using the process fluid density when the head is actually a manometer-fluid height. Forgetting that head in metres is not pressure until multiplied by rho g.

When converting 12 m of water head into the pressure delivered at a pump suction line, Hydrostatic head is used to calculate Pressure Difference from Fluid Density, Gravitational Acceleration, and Pressure Head. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

The head must be a vertical height. Use the density of the fluid that provides the head. Pressure head and pressure are proportional only when density and gravity are fixed.

References

Sources

Engineering LibreTexts, 7.9: Fluid Statics, accessed 2026-04-09
OpenStax University Physics Volume 1, Pressure in Fluids, accessed 2026-04-09
Munson, Young, Okiishi, Huebsch, and Rothmayer, Fundamentals of Fluid Mechanics, Wiley, 2013
NIST CODATA Value of the Standard Acceleration of Gravity
IUPAC Gold Book
Wikipedia article 'Hydrostatic pressure'
NIST Chemistry WebBook
Britannica

Hydrostatic head

Overview

Variables

Derivation

Start with hydrostatic pressure

Name the height as pressure head

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Hydrostatic pressure

General manometer

Differential manometer

Manometer for gas

Frequently Asked Questions

Sources