How to calculate hydrostatic force on a flat surface
Pressure acts across a surface, so a larger submerged gate can feel a greater total force. For a flat surface in still liquid, use pressure at its area centroid to find the total resultant.
What the formula is saying
With constant density, pressure increases linearly with vertical depth. Its area-average equals ρg times the centroid depth h̄. Multiply that average by area A.
Read the symbols in plain language
- ρ
- Fluid densitykg/m³
- g
- Gravitym/s²
- A
- Aream²
- h̄
- Centroid depthm
Sort out the units first
Use A in m² and centroid depth h in m. For a vertical rectangle from depth 1 m to 3 m, h̄ = 2 m, not 3 m. Pa × m² gives N.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
- ρ · Fluid density
- 1000 kg/m³
- g · Gravity
- 9.81 m/s²
- A · Area
- 2 m²
- h̄ · Centroid depth
- 2 m
Find pressure at the area centroid
This is the average gauge pressure over the flat submerged area.
(1000) × (9.81) × (2) = 19620 PaMultiply average pressure by area
Pressure is force per area; multiplying recovers force.
(19620) × (2) = 39240 N
Does this worked answer make sense?
The example gives 39.24 kN. Same depth and twice the area means twice the force, not twice the pressure.
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Use these new values. Work it out first, then check your answer.
- ρ · Fluid density
- 1000 kg/m³
- g · Gravity
- 9.81 m/s²
- A · Area
- 3 m²
- h̄ · Centroid depth
- 4 m
Find: hydrostatic force on a flat surface
A hint, not the answer
With constant density, pressure increases linearly with vertical depth. Its area-average equals ρg times the centroid depth h̄. Multiply that average by area A.
Use A in m² and centroid depth h in m. For a vertical rectangle from depth 1 m to 3 m, h̄ = 2 m, not 3 m. Pa × m² gives N.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find pressure at the area centroid
This is the average gauge pressure over the flat submerged area.
(1000) × (9.81) × (4) = 39240 PaMultiply average pressure by area
Pressure is force per area; multiplying recovers force.
(39240) × (3) = 117720 N
Avoid the common trap
The bottom-edge depth is not the centroid depth. The force generally acts at the centre of pressure, which is not the area centroid.
When this method applies — and when it does not
Flat, fully submerged surface under a hydrostatic gauge-pressure distribution. Curved surfaces and unequal gas pressures on the two sides need additional treatment.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Lesson updated: · Worked examples checked against the implemented formula; not an independent engineering certification.