Last updated: 2026-07-13

Definition

A hydraulic press is a machine that multiplies force through a confined liquid. Two pistons of different size seal a closed volume of oil or water: a small input piston and a large output piston. A force pushed onto the small piston raises the pressure of the fluid, and that pressure acts on the whole area of the large piston, producing a much larger force. The ratio of the two forces is the press's mechanical advantage.

Pascal's principle

The press works because of Pascal's principle: a pressure change applied to an enclosed, incompressible fluid is transmitted undiminished to every point of the fluid and to the walls that hold it. The pressure under the small piston is therefore the same as the pressure under the large piston:

Here $F_1$ and $A_1$ are the force and area at the input piston, and $F_2$ and $A_2$ are the force and area at the output piston.

The formula

Rearranging the equal-pressure condition gives the output force as the input force scaled by the ratio of the piston areas:

Pistons are usually circular, so each area is $A = \pi d^2 / 4$. The constant $\pi/4$ cancels in the ratio, leaving the area ratio equal to the square of the diameter ratio:

The system pressure follows from either piston. Using the input side and the circular area $A_1 = \pi d_1^2 / 4$:

Because the force scales with the diameter squared, doubling the output diameter quadruples the output force.

Worked example

A bench press has a 2 cm input piston and a 10 cm output piston. A hand pump applies 100 N to the small piston.

Diameter ratio: 10 ÷ 2 = 5

Mechanical advantage: 5² = 25

Output force: 100 × 25 = 2500 N — about the weight of a small car resting on the large piston.

System pressure: the input area is $A_1 = \pi (0.02)^2 / 4 = 3.14 \times 10^{-4}\ \text{m}^2$, so $P = 100 / (3.14 \times 10^{-4}) \approx 318\ \text{kPa}$ (about 3.1 atmospheres). The same 318 kPa presses on the large piston, whose area is 25 times bigger, giving the 2500 N output.

Force is multiplied, work is not

A hydraulic press does not create energy. The fluid that leaves the space under the small piston has to appear under the large piston, and the large piston has 25 times the area, so it moves 25 times less far. Push the small piston down 25 cm and the large piston rises just 1 cm. Multiply force by distance on each side and the work matches:

This is the same bargain offered by a lever, a screw or a block and tackle: a large force over a short distance in exchange for a small force over a long distance. Friction and fluid leakage mean the real output is always a little below the ideal figure.

Where hydraulic presses are used

Assumptions and limits

The formulas assume an incompressible fluid with no leakage, rigid cylinder walls, and pistons that are free of friction. They also ignore any height difference between the two pistons: a tall column of fluid adds its own hydrostatic pressure ($\rho g h$), which is usually small next to the working pressure but matters in precise work. Real presses reach a relief-valve limit — a maximum pressure the pump and seals can hold — so the output force cannot grow without bound. Treat the calculated output as the ideal ceiling and expect a modest loss in practice.

Notes

The two diameters must be entered in the same unit, since only their ratio enters the force and mechanical-advantage results; the ratio itself is dimensionless. The output force is reported in the same unit as the input force. Both diameters must be greater than zero for the areas — and therefore the pressure — to be defined.