Pressure, Force and Area

Pressure tells you how hard something presses on a surface. A force spread over a big area gives a small pressure. The same force on a small area gives a big pressure. That is why a drawing pin, with its sharp point, goes into a board so easily.

In the exam the formula for pressure is given in the question. You are told two of pressure, force and area, and asked to work out the third, usually for 2 or 3 marks. Pressure questions do not come up on every paper, and they are usually near the end.

Often the area is not given straight away. You first work out the area of the face touching the floor, from its length and width, or from a volume and a height.

Force is measured in newtons, written N. The force an object exerts on the floor is its weight: a bag of sugar exerts a force of about 10 newtons.

Pressure is the force on each square centimetre, or each square metre, of the area:

\text{pressure}=\frac{\text{force}}{\text{area}}

The units of pressure come from the units you divide. Newtons divided by cm² give newtons per square centimetre, written newtons/cm² or N/cm². Newtons divided by m² give newtons/m², or N/m².

For example, 60 newtons on an area of 20 cm² is 60\div20=3 newtons/cm². Each square centimetre takes 3 newtons.

If you know the pressure and the area, multiply them to get the force: 3 newtons on each of 20 cm² is 3\times20=60 newtons in all. Divide the force by the pressure to get the area.

\text{force}=\text{pressure}\times\text{area}

\text{area}=\frac{\text{force}}{\text{pressure}}

Cover the one you want to find. Covering F leaves P and A side by side, so multiply. Covering P leaves F over A, so divide force by area. Covering A leaves F over P, so divide force by pressure.

The force is the only one you find by multiplying. To find a pressure or an area, you divide.

The area in the formula is only the area touching the surface: the face of a box on the floor, or the end of a post on the ground. It is not the total surface area, and it is not the volume. A table on 4 legs, each with 5 cm² touching the floor, has 4\times5=20 cm² in contact.

For a rectangle, multiply the length by the width. A base 5 m by 3 m has area 5\times3=15 m², not 5+3=8 m². Don't square the numbers: the ² in m² is part of the unit.

A cube has 6 faces, all the same size, and it stands on just one of them. So the area you work out from the force and the pressure is the area of one face, not the whole cube.

Some questions give the volume and the height of a prism, such as a cuboid or a cylinder, standing on one end. The end touching the floor is its cross-section.

The volume of a prism is the area of its end times its height, so:

\text{area of the end}=\frac{\text{volume}}{\text{height}}

For a cylinder you don't need the radius. Just divide the volume by the height.

For example, a post with volume 1050 cm³ and height 35 cm stands on an end of area 1050\div35=30 cm².

• Multiplying when you should divide. Only the force is found by multiplying; a pressure or an area is found by dividing.
• Adding the lengths to find an area. Multiply the length by the width: a base 5 m by 3 m has area 15 m², not 8 m².
• Squaring the numbers because the unit has a ². The ² belongs to the unit, so 2.5 newtons/cm² means 2.5, not 2.5^2
• Using the wrong area. Use only the face touching the surface, not the total surface area or the volume; for a prism on its end, divide the volume by the height.
• Forgetting the other faces of a cube. The area you find from the force and the pressure is one face; a cube has 6 faces the same, so its total surface area is 6 times that.
• Slips when dividing. Check by multiplying back: pressure times area should give the force.