## What you need to know

**Area** is a measure of the size of a surface and is usually measured in square centimetres () or square metres (). **Force **is a measure of how much push or pull is being exerted onto an object and is usually measured in Newtons (N). **Pressure** is a measure of how much force is applied over a given area of an object, so it is calculated by dividing the amount of force being applied by the area over which its being applied.

As a result, the units we used to measure pressure are **compound units **(for more information, see here (https://mathsmadeeasy.co.uk/gcse-maths-revision/conversions-gcse-revision-and-worksheets/)). The unit of choice will depend on what units are used to measure the values you’re given. If, in a particular question, area is measured in square metres and force is measured in Newtons, then the pressure we calculate would be measured “Newtons per square metre”, which is denoted either by or . If the starting units were different, then the resulting unit of pressure would be different. Let’s see an example.

**Example: **A force of 150N is being applied over an area measuring . Calculate the pressure on the object.

So, pressure is equal to the force divided by the area, so this will be our calculation, but what will the unit of our answer be? In the question, the force is given in Newtons and the area in square centimetres, so the pressure should be in **Newtons per square centimetre**, which is denoted like .

This is one of the more common units of pressure, but it does vary, so watch out for the units given in the question. In general, the units of pressure will be in the form “[units of force] per [units of volume]”. On the bright side, force is almost always in Newtons.

On top of calculating pressure, you will also be expected to be able to rearrange this formula and use it to find force (when given the pressure and area) and area (when given the pressure and force). From now on, let be pressure, be area, and be force. Then, our original equation looks like:

If we multiply both sides of the equation by , then swap the left-hand side and right-hand side, we get

So, we see that we can calculate force by multiplying the pressure by the area. Furthermore, if we then divide both sides by and swap the left-hand side and right-hand side, we get

Therefore, we can calculate area by dividing the force by the pressure. It’s good practice rearranging this formula to get it in the form that you want, but a quick way to remember how to calculate one of these values using the other two is to refer to the triangle below.

The way to use this triangle is as follows. Take your finger and cover up the letter which represents the thing that you’re trying to calculate. Then, the triangle will tell you what to do with the other two quantities to get the value you want.

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For example, if we want to calculate the pressure, then we construct this triangle and cover up the (since that’s what we want). Then, we see that what’s left over is “ over ”, or in other words, divided by will give us the pressure. Learning this triangle and how to use it makes your life a lot easier.

**Example: **A woman is applying 300\text{ N/m}^2 of pressure onto a door with her hand. Her hand has area 0.02\text{ m}^2. Work out the force being applied.

We’re looking for force, so, constructing the triangle and covering up the , we get

Therefore, to calculate the force we must multiply the pressure by the area. So

**Note: **this question works fine because all the units match up. If, for example, pressure was given in ‘Newtons per __square metre__’ but area was given in ‘__square centimetres__’, we couldn’t work out the force straight away. We would first have to either convert the area to ‘__square metres__’ so it matched the pressure, or convert the pressure into ‘__Newtons per square centimetre__’ so it would match the area.

### Example Questions

1) A force of 185.6N is being applied to a square of side-length 3m. Work out the pressure on the square to 3sf.

We are looking for pressure, and we can see by covering up p in the triangle above that we need to divide F by A to get what we want. Before we can do this calculation however, we need to actually work out the area – a square with side-length 3m has an area equal to

3^2=9\text{ m}^2.

So, now we have the area and the force, we get

\text{pressure }=\dfrac{F}{A}=\dfrac{185.6}{9}=20.6222...=20.6\text{ N/m}^{2}\text{ (3sf)}

The units must be \text{N/m}^2 since the units used in the question are N and \text{m}^2.

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2) A man is standing on one foot, and this foot is applying 740N to the ground. As a result, this foot is exerting 2,312.5\text{ N/m}^2 of pressure onto the ground. What is the surface area of the bottom of his foot?

We are looking for area, and we can see by covering up A in the triangle above that we need divide F by p to get what we want. So, we get

\text{area }=\dfrac{F}{p}=\dfrac{2,312.5}{740}=3.125\text{ m}^2

The units must \text{m}^2 since the units used in the question are \text{N/m}^2.

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## Pressure Force Area Worksheets and Revision

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