Pressure force area

Here you will learn about pressure force and area, including what they are and how they are related to one another.

Students will first learn about pressure force and area as part of algebra in high school.

What are pressure, force, and area?

Pressure, force, and area are physical properties.

Area

Area is a measure of the size of space a flat shape takes up. The SI unit for area is the square meter (m^2) although you can use other units including square centimeters (cm^2), square millimeters (mm^2), square inches (in^2), etc.

Area is inversely proportional to the pressure exerted; for a constant force, if the area in which the force is being applied increases, the pressure exerted across that area decreases.

You can therefore state that A\propto\cfrac{1}{P} for the area, A and the pressure, P.

The constant of proportionality is the force, F.

Note: When you consider the area of an object, you are referring to the area of contact of the object with the floor, table, ground, wall or shelf, etc.

Force

Force is the energy attributed to a movement or physical action. Force is measured in the standard units of force, Newtons (N).

Newton’s Second Law of Motion states that force is equal to the mass of an object, multiplied by the acceleration, or F=MA. This is why weight and mass are not the same.

Mass is a measure of how much matter there is in an object (usually measured in kilograms), whereas weight is the size of the pull of gravity on the object (measured in Newtons).

The acceleration due to gravity on the surface of the Earth is 9.807{~m/s^2} and so an average human with a mass of 62{~kg} would have a weight of 62 \times 9.807=608{~N}.

The force acting on an object is directly proportional to the pressure applied over a constant area. The greater the pressure, the greater the force for a constant area and vice versa.

You can therefore state that F \propto P for the force F and the pressure P. The constant of proportionality is the area, A.

See also: Directly proportional

The forces used for this topic act on an object that is in contact with the floor, table, wall, shelf, etc. This means that the force will be perpendicular to the surface for all questions.

Pressure

Pressure is a compound measure. Pressure is defined as the force per unit area. The standard unit of pressure is Pascals (Pa) where 1 \ Pa=1 \ N / {m^2}.

You can also measure pressure using a Bar, where 1 Bar = 100,000 Pa = 100,000 N/ {m^{2}}.

Imagine an elephant standing on 4 legs. The pressure applied to the ground is distributed over a wider area than if the elephant were standing on one leg.

This is because the force remains constant (the weight of the elephant), but the area in which the force is applied has decreased, so the pressure increases. This is an example of inverse proportion.

As pressure is inversely proportional to the area, you can state that P\propto\cfrac{1}{A} for the pressure, P, and the area, A. The constant of proportionality is the force, F.

See also: Inversely proportional

To calculate either the pressure, force, or area of an object, you use the pressure formula (or pressure equation):

\text{Pressure}=\cfrac{\text{Force}}{\text{Area}}

Pressure triangle

This can be written as a formula triangle (sometimes called the pressure triangle).

You can circle what you are trying to find and the formula triangle tells how to calculate the unknown property.

Step-by-step guide: Pressure formula

What are pressure, force, and area?

Common Core State Standards

How does this relate to high school math?

• High School – Algebra – Creating Equations (HS.A.CED.A.1)
  Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

• High School – Algebra – Creating Equations (HS.A.CED.A.2)
  Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

• High School – Algebra – Creating Equations (HS.A.CED.A.4)
  Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

How to calculate pressure, force, or area

In order to calculate pressure, force, or area using the formula triangle:

1. Draw the triangle and circle the required property.
2. Substitute values for the other two properties and complete the calculation.
3. Write down the solution, including the units.

Pressure force area examples

Example 1: calculating pressure (Pa)

A force of 800{~N} acts on an area of 200{~m^2}. Calculate the pressure in Pascals.

1. Draw the triangle and circle the required property.

To calculate the pressure, you would need to divide the force by the area.

2Substitute values for the other two properties and complete the calculation.

Substituting F=800{~N} and A=200{~m^2}, you have:

3Write down the solution, including the units.

Pressure = 4 \ N/{m^{2}}=4 \ Pa.

Example 2: calculating pressure (Bar)

A force of 6.5 \times 10^6 N acts on an area of 0.2{~m^2}. Calculate the pressure in Bar.

Draw the triangle and circle the required property.

To calculate the pressure, you would need to divide the force by the area.

Substitute values for the other two properties and complete the calculation.

Converting 6.5 \times 10^6 to an ordinary number, you get F=6500000{~N}.

P=6500000\div{0.2}=32500000

Example 3: calculating area (m²)

A force of 200{~N} exerts a pressure of 40{~N/{m^2}}. Calculate the area of the surface that the force is being applied to in square meters.

Draw the triangle and circle the required property.

To calculate the area, you would need to divide the force by the pressure.

Substitute values for the other two properties and complete the calculation.

Substituting F=200{~N} and P=40 \ N/{m^{2}}, you have:

A=200\div{40}=5

Example 4: calculating area (converting units of pressure)

A force of 950{~N} exerts a pressure of 53{~Pa}. Calculate the area in square meters to the nearest hundredth.

Draw the triangle and circle the required property.

To calculate the area, you would need to divide the force by the pressure.

Substitute values for the other two properties and complete the calculation.

As you need the units for the area to be in square meters, you need to convert the pressure from Pascals into N/m^2. As 1 Pa=1N/m^2, you can simply state that our pressure P=53N/m^2.

Substituting F=950N and P=53N/{m^2} into the pressure triangle, you have

A=950\div{53}=17.9245283\dots

Example 5: calculating force (N)

A box has a cross-sectional area of 1.2{~m^2} and exerts a pressure of 60{~N/{m^2}} on the ground. Calculate the force that the box exerts on the ground, in Newtons.

Draw the triangle and circle the required property.

To calculate the force, you would need to multiply the pressure by the area.

Substitute values for the other two properties and complete the calculation.

As P=60 N/{m^2} and A=1.2{m^2}, substituting these into the formula triangle you get:

F=60\times{1.2}=72

Example 6: calculating force (converting units)

A pressure of 35 \ N/{m^2} is exerted on an area of 40,000{~cm^2}. Calculate the force in Newtons.

Draw the triangle and circle the required property.

To calculate the force, you would need to multiply the pressure by the area.

Substitute values for the other two properties and complete the calculation.

The area is in square centimeters, so you need to divide by 100^2 to find the area in square meters.

40000\div{100}^{2}=4

Substituting P=35 \ N/{m^2} and A=4{~m^2} into the pressure triangle, you have

F=35\times{4}=140

Example 7: calculating pressure given the mass

An object has a mass of 2.4{~kg} and makes contact with the surface of the Earth with an area of 14.9{~m^2}. Given that the acceleration due to gravity g=9.807 \ m/{s^2}, calculate the pressure exerted on the Earth by the object to the nearest hundredth. State the units in your answer.

Draw the triangle and circle the required property.

Before you use the formula triangle, you need to calculate the force exerted on the Earth by the object. This can be found by multiplying the mass by the acceleration due to gravity (F=MA).

F=2.4\times{9.807}=23.5368 \ N

As you want to determine the pressure, you need to divide the force by the area:

Substitute values for the other two properties and complete the calculation.

Now you have F=23.5368{~N} and A=14.9{~m^2}.

Substituting these values into the pressure triangle, you have:

P=23.5368\div{14.9}=1.579651007\dots

Teaching tips for pressure force area

• Encourage students to rearrange the formula to solve for force or area, so they can see how each variable impacts the others directly.

• Explain that pressure applies differently in solids and fluids. In a solid, force is typically distributed over the contact area in one direction. In contrast, fluids (liquids and gases) apply pressure evenly in all directions, which is why pressure gauges can measure uniform fluid pressure.

• Introduce force as a vector quantity, with both magnitude and direction (example, a push of 10 Newtons applied upward or sideways). In contrast, explain that pressure is a scalar quantity, having only magnitude—when force is spread over an area, it becomes uniformly distributed as pressure, with no specific direction.

• Show real-world examples like hydrostatic pressure in dams, where water pressure against the dam wall increases with depth. Use this to explain the need for dam walls to be thicker at the base, highlighting the interaction between total pressure, force, and area.

Easy mistakes to make

• Thinking that area is the total surface area of the object
  The area is the amount of surface with which the object is in contact. This will usually be one face of an object, or given as the cross-sectional area.

• Mixing up the placement of the letters in the triangle
  Formula triangles have limited use and if the letters are incorrectly written in the wrong place, the subsequent calculation will be incorrect.
  
  To overcome this, the standard units of pressure are N/{m^2} which is a force (N), divided by an area (m^2) and so P=F \div A.

Related compound measure lessons

• Speed distance time
• Speed formula
• Average speed formula
• Density formula
• Density mass volume triangle
• Mass from density and volume
• Flow rate
• Population density formula

Did you know?

• Air pressure (atmospheric pressure) can change depending on where you are in the world. If you are up a mountain, the air pressure is less than when you are at sea level.

• Each time a particle comes into contact with a wall, it exerts a force on the wall. If you increase the number of particles within the container (such as an inflating balloon), the frequency of collisions increases, and so does the pressure of the gas inside.
  
  At high pressure, the balloon may not be able to withstand the increased amount of force on its surface, and so it bursts.

The next lessons are

• Ratio
• Proportion
• Converting fractions decimals and percentages

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