Math resources Ratio and proportion

Compound measures

Pressure formula

# Pressure formula

Here you will learn about the pressure formula, including how to use the formula. Students will first learn about the pressure formula as part of ratio and proportionality in 7 th grade.

## What is the pressure formula?

The pressure formula is a formula connecting pressure, area, and force.

Pressure is a measure of force per unit area. Weight is calculated by multiplying mass by the acceleration of gravity so heavier objects exert more pressure.

To calculate the pressure, area or force applied to an object, you need to use their intrinsic relationship. Pressure is calculated by dividing the amount of force by the area.

The pressure formula (or pressure equation) is therefore

\text{Pressure}=\cfrac{\text{Force}}{\text{Area}} or P=\cfrac{F}{A}

Pressure is a compound measure made from force and area. Pressure equals the force per unit area. Pressure is measured using pressure gauges.

Force is the energy attributed to a movement or a physical action. The SI unit for force is Newtons (N).

Area is the amount of a surface on which the force is applied to. The standard unit for area is square meters (m^{2}).

The pressure formula can be used in a formula triangle where it can be adapted to calculate force and area.

To find pressure \hspace{1.3cm} To find force \hspace{1.5cm} To find area

\hspace{1.2cm} P=\cfrac{F}{A} \hspace{3cm} F=PA \hspace{2.5cm} A=FP

The units of force are Newtons (N) and the derived SI units of area are square meters (m^{2}). So the units of pressure are Newtons per square meter (N/m^{2}).

However, the standard unit of pressure is Pascals (Pa). \, 1 Pascal is equal to 1 \, N/m^{2} and 100,000 \, Pa=1 Bar (another unit of pressure).

Air pressure (or atmospheric pressure) can change depending on where you are on planet Earth. If you are on a mountain, the air pressure is less than when you are at sea level. The air pressure at sea level is approximately 14.7 pounds per square inch or 1.01325 Bar.

## Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – Ratio and Proportionality (7.RP.A.2)
Recognize and represent proportional relationships between quantities.

## How to calculate the pressure, force or area using the pressure formula

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

1. Write the formula with the correct subject.
2. Substitute known values into the formula and do the calculation.
3. Write down the solution including the units.

## Pressure formula examples

### Example 1: calculating pressure (N/m²)

A force of 600 \, N acts on an area of 20 \, m^{2}. Calculate the pressure.

1. Write the formula with the correct subject.

As P=\cfrac{F}{A} and you want to calculate the pressure, you do not need to rearrange the formula, so you have,

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

2Substitute known values into the formula and do the calculation.

\text{Pressure}=\cfrac{\text{Force}}{\text{Area}}=\cfrac{600}{20}=30

3Write down the solution including the units.

Pressure is 30 \, N/m^{2}.

The solution can also be written in Pascals, 30 \, Pa.

### Example 2: calculating pressure (N/m²)

A force of 980 \, N acts on an area of 120 \, m^{2}. Calculate the pressure. Give your answer correct to the nearest hundredth.

Write the formula with the correct subject.

Substitute known values into the formula and do the calculation.

Write down the solution including the units.

### Example 3: calculating area (m²)

A force of 560 \, N exerts a pressure of 70 \, N/m^{2}. Calculate the area of the surface that the force is applied to.

Write the formula with the correct subject.

Substitute known values into the formula and do the calculation.

Write down the solution including the units.

### Example 4: calculating area (m²)

A force of 460 \, N exerts a pressure of 18 \, N/m^{2} onto a flat surface. Calculate the area of the surface that the pressure is being exerted on. Give the answer correct to the nearest tenth.

Write the formula with the correct subject.

Substitute known values into the formula and do the calculation.

Write down the solution including the units.

### Example 5: calculating force (N)

A pressure of 40 \, N/m^{2} is exerted on an area of 6 \, m^{2}. Calculate the force.

Write the formula with the correct subject.

Substitute known values into the formula and do the calculation.

Write down the solution including the units.

### Example 6: calculating the force (N)

A pressure of 83 \, N/m^{2} is exerted on an area of 53.2 \, m^{2}. Calculate the force.

Write the formula with the correct subject.

Substitute known values into the formula and do the calculation.

Write down the solution including the units.

### Example 7: converting units of area

A pressure of 29 \, N/m^{2} is exerted on an area of 20000 \, cm^{2}. Calculate the force applied to the area.

Write the formula with the correct subject.

Substitute known values into the formula and do the calculation.

Write down the solution including the units.

### Teaching tips for pressure formula

• Provide practice problems where students calculate pressure given force and area, and vice versa. Include units to reinforce the importance of proper unit usage.

• Use visual aids and experiments, such as standing on one foot versus both feet, to demonstrate how distributing force over a larger surface area affects pressure.

• Introduce the Ideal Gas Law PV=nRT to explain gas pressure. Discuss how gas pressure increases with temperature and decreases with volume.
• In thermodynamics, the ideal gas law states that the gas constant R is equal to the pressure times the volume of a gas, divided by the number of moles and temperature of the gas.
R=8.3145J/mol/K (where J= Joules, mol= Moles, K= Kelvin)

• Introduce the concept of fluid pressure and explore topics such as Pascal’s principle, buoyancy, and Bernoulli’s principle. Bernoulli’s principle states that in a fluid flow, an increase in velocity results in a decrease in pressure. This principle is crucial in explaining phenomena like the lift on airplane wings and the operation of Venturi meters.

### Easy mistakes to make

• Using incorrect units
As pressure is a measure of force per unit area (N/m^{2}), the units for force will be given (N), as well as the units for area (m^{2}). Use these to determine the units of your answer. If you are given the pressure in Pascals, 1 \, Pa = 1N/m^{2} and so you will need to remember this to determine the units of your solution.

• Not fully reading the question
Check that you give the answer in the right form. You may have an answer which needs to be rounded to a specific decimal place.

• Not converting units of area correctly
To convert centimeters to meters, you divide by 100 as 1 \,m=100 \, cm. To convert square centimeters to square meters, you divide by 100^{2}=10,000 as 1 \, m^{2} = 10,000 \, cm^{2}.

• Confusing scalar and vector
Area is a scalar quantity as it has size. Force is a vector quantity as it has size and magnitude. Pressure is a scalar quantity too.

### Practice pressure formula questions

1. A force of 600 \, N acts on an area of 30 \, m^{2}. Calculate the pressure.

20 \, N/m^{2}

0.2 \, N/m^{2}

2 \, N/m^{2}

200 \, N/m^{2}
\text{Pressure}=\cfrac{\text{Force}}{\text{Area}}=\cfrac{600}{30}=20 \, N/m^{2}

2. A force of 623 \, N acts on an area of 452 \, m^{2}. Calculate the pressure. Give your answer correct to the nearest hundredth.

29.3 \, N/m^{2}

282000 \, N/m^{2}

0.726 \, N/m^{2}

1.38 \, N/m^{2}

\text{Pressure}=\cfrac{\text{Force}}{\text{Area}}=\cfrac{623}{452}=1.3783…=1.38N/m^{2}   (nearest hundredth)

3. A force of 800 \, N exerts a pressure of 20 \, N/m^{2}. Calculate the area that the force is applied to.

400 \, m^2

40 \, m^2

30 \, m^2

16 \, m^2
\text{Area}=\cfrac{\text{Force}}{\text{Pressure}}=\cfrac{800}{20}=40 \, m^2

4. A force of 463 \, N exerts a pressure of 9.8 \, N/m^{2}. Calculate the area that the force is applied to, correct to the nearest hundredth.

4537.40 \, m^2

47.24 \, m^2

0.02 \, m^2

4.82 \, m^2

\text{Area}=\cfrac{\text{Force}}{\text{Pressure}}=\cfrac{463}{9.8}=47.24489796…=47.24 \, m^{2} (nearest hundredth)

5. A pressure of 25 \, N/m^{2} is exerted on an area of 6 \, m^{2}. Calculate the force.

4.2 \, N

0.69 \, N

150 \, N

900 \, N
\text{Force}=\text{Pressure}\times \text{Area}=25\times 6=150 \, N

6. A pressure of 89 \, N/m^{2} is exerted on an area of 17.3 \, m^{2}. Calculate the force. State the units in your answer.

1539.7 \, g

15.39.7 \, kg

1539.7 \, Pa

1539.7 \, N
\text{Force}=\text{Pressure}\times\text{Area}=89\times{17.3}=1539.7 \, N

## Pressure formula FAQs

What is the pressure formula?

The pressure formula is P=\cfrac{F}{A}, where pressure P, is calculated by dividing the amount of force F by the area A.

What are the units of pressure?

The SI unit of pressure is the Pascal (Pa). One Pascal is equivalent to one Newton per square meter.

1 \mathrm{~Pa}=1 \mathrm{~N}/m^{2}

What is the difference between gauge pressure and absolute pressure?

Gauge pressure is the pressure relative to atmospheric pressure. Absolute pressure is the total pressure, including atmospheric pressure. Absolute pressure = Gauge pressure + Atmospheric pressure.

What is the difference between absolute pressure and partial pressure?

Absolute pressure is the pressure that exists in a space with no matter, such as in a perfect vacuum. Partial pressure is the pressure exerted within a mixture by an individual gas (such as the partial pressure of nitrogen gas in the air at sea level is 593 \, mmHg or 79058.76 \, Pa).

What is hydrostatic pressure?

In fluid mechanics, hydrostatic pressure (fluid pressure or water pressure which occurs in open conditions such as swimming pools or the ocean) is the pressure in a liquid at a given depth.

The hydrostatic pressure can be calculated using the formula P=ρgh where P represents the pressure, ρ the greek letter rho is the density of the liquid, g is acceleration due to gravity for the height h (or depth) of the liquid.

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