Hydraulic press: diameter is not area

A tempting wrong shortcut

Suppose a circular output piston has three times the input diameter. Does that give three times the ideal output force? It is a tempting shortcut, but the face area tells a different story. Today we will find the ratio and check what it predicts.

Look at two dimensions

A circle with three times the diameter is also three times as tall in this end-on view. Its area becomes nine times as large. In the circle formula, area depends on diameter squared. The factor involving pi is the same for both faces.

Calculate the ratio

Our input diameter is two centimeters and our output diameter is six centimeters. Divide six by two to get three, then square three to get nine. This gives the area ratio. The diagram uses the same scale for both circles so you can inspect the difference.

Pause before multiplying

Assume equal piston heights, negligible friction and piston weight, and the same outside pressure. The net input force is seven newtons. Pause and find the ideal output force. Be careful to use the area ratio, not just the diameter ratio.

Check force and travel

Seven times nine gives sixty three newtons. With no leaks or compression, nine centimeters of input travel would give one centimeter of output travel. The force rises by the same factor that the travel falls. The energy budget still has to balance.

Pressure needs careful units

To calculate pressure in pascals, divide newtons by square meters. Three square centimeters is zero point zero zero zero three square meters. Twelve newtons over that area gives forty thousand pascals, or forty kilopascals. Converting length alone would give the wrong area conversion.

The height surprise

Fun fact: points at different depths in still liquid can have different pressures. Pascal's principle concerns a transmitted pressure change, not a promise that every pressure reading everywhere is identical. Our simple equal-pressure force ratio needs the same-height assumption, or an appropriate height correction.

Keep your assumptions

You can now square a diameter ratio and predict both force and travel. These calculations explain an ideal model; they are not machine-design instructions. Continue to the worksheet to test the squared ratio, compare travel, and see what happens when both diameters change.