Science
What is pressure in physics, and what is the pressure formula?
Quick answer
Pressure is force spread over an area, p = F ÷ A, measured in pascals (1 Pa = 1 N/m²). In a liquid it grows with depth, p = ρgh.
In one sentence
Pressure tells you how concentrated a force is: how much force pushes on each square metre of a surface.
Explained
Walk across deep snow in ordinary boots and you sink. Strap on snowshoes and you stay on top, though you weigh the same: your weight is now spread over a much bigger area. That is pressure:
pressure = force ÷ area, p = F / A
Force is measured in newtons (N) and area in square metres (m²), so pressure is in newtons per square metre, N/m². This unit has its own name, the pascal (Pa): 1 Pa = 1 N/m². A pascal is tiny (about the pressure of a sheet of paper lying on a table), so you will often see kilopascals: 1 kPa = 1000 Pa. For small areas, N/cm² is also used: 1 N/cm² = 10,000 Pa.
The formula comes from the meaning: if 600 N is spread over 3 m², each square metre carries 600 ÷ 3 = 200 N, so the pressure is 200 Pa. Rearranged, F = p × A and A = F ÷ p.
elephant
shoe
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Big pressure, small pressure. To make pressure big, push on a small area: knives, needles, nails, drawing pins and ice-skate blades all have tiny contact areas. To make pressure small, spread the force over a big area: snowshoes, wide tractor tyres, the broad feet of camels and elephants, and wide straps on a heavy bag. Pressure is linked closely to forces and area.
Pressure in liquids. Water pushes on everything in it, in all directions, and the deeper you go, the more water is above you pressing down. The pressure caused by a liquid is
p = ρ × g × h
where ρ is the liquid's density (kg/m³), g is the gravitational field strength (9.8 N/kg on Earth) and h is the depth (m). In fresh water, every 10 m of depth adds 1000 × 9.8 × 10 = 98,000 Pa, almost one atmosphere. It depends only on depth and density, not on the container's shape: that is why dams are thicker at the bottom.
Atmospheric pressure. The air above us also has weight. At sea level it presses with about 101,000 Pa (101 kPa), called 1 atmosphere. Weather forecasts give it in hectopascals: 1013 hPa is typical. We do not feel it because it pushes from every side and our bodies push back. Higher up there is less air above, so the pressure falls: crisp packets puff up on a mountain.
Gauge and absolute pressure. A tyre gauge measures the pressure above atmospheric pressure (gauge pressure). The absolute pressure is gauge pressure + atmospheric pressure.
Pressure in gases. Gas particles move fast and keep hitting the walls of their container. Each hit is a tiny push, and together they make the gas pressure. Squash a gas into a smaller volume and the particles hit the walls more often, so the pressure rises. For a fixed amount of gas at a constant temperature, p₁V₁ = p₂V₂ (Boyle's law): halve the volume and the pressure doubles. Heating a gas in a closed container also raises its pressure, because the particles move faster.
Hydraulics. Liquids barely squash, so a pressure applied in one place spreads through the whole liquid (Pascal's principle). A small force on a small piston gives a big force on a big piston: that is how car brakes and hydraulic jacks work.
Examples
| Situation | Force | Area | Pressure |
|---|---|---|---|
| Person standing on two feet | 600 N | 0.03 m² | 20,000 Pa |
| Same person on one foot | 600 N | 0.015 m² | 40,000 Pa |
| Same person on snowshoes | 600 N | 0.3 m² | 2,000 Pa |
| Thumb on a drawing pin's point | 20 N | 0.0000001 m² (0.1 mm²) | 200,000,000 Pa |
| Water at a depth of 10 m | — | — | 98,000 Pa (p = ρgh) |
Worked example: a box weighs 800 N and its base is 0.5 m × 0.4 m. Area = 0.2 m². Pressure = 800 ÷ 0.2 = 4,000 Pa (4 kPa).
Common mistakes
- Converting cm² to m² wrongly. 1 m² = 100 cm × 100 cm = 10,000 cm², not 100. So 150 cm² = 0.015 m².
- Using mass instead of weight. The force must be in newtons. A 50 kg box has a weight of 50 × 9.8 = 490 N.
- Thinking a wider container means more pressure. At the same depth, a narrow tube and a wide lake give the same liquid pressure.
- Forgetting the air. p = ρgh gives the pressure from the liquid only. For the total (absolute) pressure underwater, add atmospheric pressure.
Step by step
Step by step
- Find the forceIn newtons. If you are given a mass, multiply by g (9.8 N/kg) to get the weight.
- Find the areaIn m². Convert first if needed: divide cm² by 10,000.
- DividePressure = force ÷ area.
- Add the unitN/m² is the same as Pa. Large answers can be given in kPa (÷ 1000).
- Sense-checkA smaller area should give a bigger pressure. Does your answer fit?
For a liquid, multiply instead: p = density × 9.8 × depth. For a gas squeezed at a constant temperature, use p₁V₁ = p₂V₂.
Practice questions
Pressure calculations
Try each one, then check your answer.
- A force of 100 N acts on an area of 2 m². Find the pressure.
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100 ÷ 2 = 50 Pa - A force of 600 N acts on 0.03 m². Find the pressure.
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600 ÷ 0.03 = 20,000 Pa (20 kPa) - A force of 50 N acts on 10 cm². Find the pressure in N/cm² and in Pa.
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50 ÷ 10 = 5 N/cm²; 10 cm² = 0.001 m², so 50 ÷ 0.001 = 50,000 Pa - A pressure of 2000 Pa acts on 3 m². What is the force?
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F = p × A = 2000 × 3 = 6000 N - A force of 900 N gives a pressure of 30,000 Pa. What is the area?
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A = F ÷ p = 900 ÷ 30,000 = 0.03 m² - Convert 25 cm² into m².
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25 ÷ 10,000 = 0.0025 m² - A child weighing 450 N stands on feet with a total area of 0.025 m². Find the pressure.
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450 ÷ 0.025 = 18,000 Pa - A 50 kg box has a base of 0.5 m × 0.4 m. Find the pressure on the floor (g = 9.8 N/kg).
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Weight = 50 × 9.8 = 490 N; area = 0.2 m²; 490 ÷ 0.2 = 2450 Pa - Find the water pressure at a depth of 2 m (ρ = 1000 kg/m³, g = 9.8 N/kg).
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1000 × 9.8 × 2 = 19,600 Pa - Find the pressure from seawater (ρ = 1030 kg/m³) at a depth of 10 m.
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1030 × 9.8 × 10 = 100,940 Pa, about 101 kPa - At a depth of 10 m in fresh water the water adds 98,000 Pa. With atmospheric pressure at 101,000 Pa, what is the total pressure?
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98,000 + 101,000 = 199,000 Pa, about twice the pressure at the surface - At what depth in fresh water is the water pressure 49,000 Pa?
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h = p ÷ (ρg) = 49,000 ÷ 9,800 = 5 m - A gas at 100 kPa has a volume of 6 L. It is squashed to 2 L at the same temperature. Find the new pressure.
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p₂ = 100 × 6 ÷ 2 = 300 kPa - A gas at 200 kPa fills 3 L. It expands at the same temperature until the pressure is 150 kPa. Find the new volume.
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V₂ = 200 × 3 ÷ 150 = 4 L - In a hydraulic jack, 50 N pushes on a piston of 0.01 m². The big piston has an area of 0.5 m². What force does it give?
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p = 50 ÷ 0.01 = 5000 Pa; F = 5000 × 0.5 = 2500 N - Convert 1013 hPa into Pa.
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1 hPa = 100 Pa, so 1013 × 100 = 101,300 Pa
Word problems
Pressure in real life
Try each one, then check your answer.
- A 5000 kg elephant stands on four feet, each 0.1 m². A 60 kg person puts all her weight on one stiletto heel of 1 cm². Who presses harder on the floor? (g = 9.8 N/kg)
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Elephant: 49,000 ÷ 0.4 = 122,500 Pa. Heel: 588 ÷ 0.0001 = 5,880,000 Pa. The heel, about 48 times more. - A 70 kg hiker wears boots with a total area of 0.05 m², then snowshoes with 0.25 m². Find both pressures.
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Weight = 686 N. Boots: 13,720 Pa; snowshoes: 2,744 Pa, five times less, so she sinks far less. - You push a drawing pin with 20 N. The head is 1 cm² and the point 0.1 mm². Find the pressure on your thumb and at the point.
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Thumb: 20 ÷ 0.0001 = 200,000 Pa; point: 20 ÷ 0.0000001 = 200,000,000 Pa, 1000 times more. - A diver is 30 m deep in seawater (ρ = 1030 kg/m³). Find the water pressure, and the total pressure if the air adds 101 kPa.
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1030 × 9.8 × 30 = 302,820 Pa ≈ 303 kPa from the water; total ≈ 404 kPa, about 4 atmospheres. - A sealed syringe holds 60 mL of air at 100 kPa. You push the plunger in to 20 mL without the temperature changing. Find the pressure.
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100 × 60 ÷ 20 = 300 kPa - A 4000 kg tractor has wide tyres touching the field over 0.8 m² in total. Find the pressure.
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Weight = 39,200 N; 39,200 ÷ 0.8 = 49,000 Pa
Questions people ask
What is the formula for pressure?
p = F ÷ A: pressure equals force in newtons divided by area in square metres. For liquids, p = ρgh (density × g × depth).
What is the SI unit of pressure?
The pascal, Pa, equal to 1 N/m². Larger units are the kilopascal (1000 Pa), the bar (100,000 Pa) and the atmosphere (about 101,325 Pa). US tyre gauges use psi: 1 psi ≈ 6.9 kPa.
What is pressure in simple words?
How hard something presses on each bit of a surface. The same push feels gentle with a flat hand and painful with a fingertip, because the fingertip's area is smaller.
How do you calculate water pressure at a depth?
Multiply the density of water (1000 kg/m³) by g (9.8 N/kg) and by the depth in metres. At 5 m: 1000 × 9.8 × 5 = 49,000 Pa.
What is the difference between gauge and absolute pressure?
Absolute pressure is measured from a complete vacuum. Gauge pressure is measured above the air around us, so absolute = gauge + atmospheric pressure.
How are pressure and volume linked in a gas?
For a fixed amount of gas at a constant temperature, pressure × volume stays the same (Boyle's law, p₁V₁ = p₂V₂). Halve the volume and the pressure doubles.
What is the dimensional formula of pressure?
[M L⁻¹ T⁻²]. Pressure is force ÷ area, and 1 Pa = 1 N/m² = 1 kg m⁻¹ s⁻².
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