Pressure in a liquid 

The pressure in a liquid also called hydrostatic pressure is the force that a liquid can exert on things. The formula to use to calculate the pressure due to a liquid is the following:

Next, we show you how we got this formula using some basic math. Consider the container you see above that contains a liquid. 

It is the weight of the liquid above that exerts some pressure on the base or bottom of the container, so the formula becomes.

 We saw in the lesson about density that

Similarly,

weight = weight density × volume

volume  = area × height

weight = weight density × area × height

Replace the weight with weight density × area × height in the formula for pressure above.

Whenever you see the same thing on top and at the bottom, it gets cancelled. As a result, the area shown in bold below will cancel.

Expanding the liquid pressure formula 

Weight density  = weight / volume

According to Newton's second law, weight  = mass × g 

g is the acceleration due to gravity.

Weight density  = (mass × g) / volume

Weight density  = (mass / volume) × g

By definition, density  = ρ = mass / volume

Therefore, weight density  = ρ × g

As already stated, pressure or liquid pressure = weight density × depth

After replacing weight density with ρ × g, we get:

Pressure or liquid pressure = ρ × g × depth

Why does the pressure in a liquid depend on the density?

The denser a liquid is, the heavier the liquid is and as a result, the more pressure it will exert.

For example, mercury is more dense than water and it is heavier. It will as a result exert more pressure. Don't believe me? Just measure the weight of the same amount of mercury and water.

Why does the pressure in a liquid depend on the depth?

Suppose you go diving. The deeper you are, the more water are on top of you. Since there are more water pressing on your body, the pressure will be more.

The pressure in a liquid does not depend on the amount of liquid

To show this, we will use formula for pressure.

The tube on the right is 4 times as large as the tube on the left.

Let w = weight of the liquid inside the tube on the left and W = weight of the liquid inside the tube on the right. 

W = 4 × w

Let a = area where the pressure is applied to the bottom of the tube on the left

Let A = area where the pressure is applied to the bottom of the tube on the right.

A = 4 × a

Since 4 is on top and at the bottom, it will cancel. As you see, the pressure is still the same.

Do you see also why shape does not matter? Check this page about Pascal's vases. You will notice that the water is at the same level in each vase regardless of the shape. It is so because the pressure is equal at the bottom of each vase. If the pressures were not equal, water will flow.