GCSE Maths: Density and pressure (Higher)
Part of the GCSE Maths units and compound measures chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Density and pressure
Density is another compound measure. It tells you how much mass is packed into each unit of volume. A block of iron and a block of wood of the same size do not have the same mass, because iron has a greater density. Density = mass ÷ volume, and the usual units are g/cm³ (grams per cubic centimetre) or kg/m³. Water has a density of 1 g/cm³, so 1 cm³ of water has a mass of 1 g. Materials with a density less than 1 g/cm³ float on water.
Cover M and you see D next to V: mass = density × volume. Cover D: density = mass ÷ volume. Cover V: volume = mass ÷ density. This works the same way as the speed triangle: only the names have changed. Keep the units matching: grams with cm³ give g/cm³, kilograms with m³ give kg/m³.
Pressure
Pressure measures how concentrated a force is. A force is a push or pull, measured in newtons (N). If you spread the same force over a bigger area the pressure is smaller. That is why snow shoes stop you sinking into snow, and why a sharp knife cuts better than a blunt one: the same force acts on a much smaller area. Pressure = force ÷ area, measured in N/m² (newtons per square metre, also called pascals, Pa) or N/cm².
The formulae are pressure = force ÷ area, force = pressure × area and area = force ÷ pressure. Again, match the units: N with m² gives N/m², and N with cm² gives N/cm².
Changing the units of density and pressure (Higher)
To change g/cm³ to kg/m³, change the mass unit (g → kg is ÷ 1000) and the volume unit (cm³ → m³ is ÷ 1 000 000). Dividing by 1000 and then dividing by 1 000 000 on the bottom makes the number 1000 times bigger overall, so 1 g/cm³ = 1000 kg/m³. Similarly 1 N/cm² = 10 000 N/m², because there are 10 000 cm² in 1 m².
Key ideas
- Density = mass ÷ volume; mass = density × volume; volume = mass ÷ density.
- Pressure = force ÷ area; force = pressure × area; area = force ÷ pressure.
- Match the units (g with cm³, kg with m³, N with m² or cm²).
- Water has density 1 g/cm³. Less than that floats, more sinks.
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (450 g) and the volume (50 cm³).
We want the density. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.density = mass ÷ volume
- Put the numbers in and calculate. Use the values with matching units.density = 450 ÷ 50 = 9
- Write the answer with its unit. The unit is g/cm³.density = 9 g/cm³
Check. Work backwards: 9 × 50 = 450 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the density (2.7 g/cm³) and the volume (40 cm³).
We want the mass. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.mass = density × volume
- Put the numbers in and calculate. Use the values with matching units.mass = 2.7 × 40 = 108
- Write the answer with its unit. The unit is g.mass = 108 g
Check. Work backwards: 108 ÷ 40 = 2.7 (the density). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (56 g) and the density (0.8 g/cm³).
We want the volume. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.volume = mass ÷ density
- Put the numbers in and calculate. Use the values with matching units.volume = 56 ÷ 0.8 = 70
- Write the answer with its unit. The unit is cm³.volume = 70 cm³
Check. Work backwards: 0.8 × 70 = 56 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (600 N) and the area (0.5 m²).
We want the pressure. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.pressure = force ÷ area
- Put the numbers in and calculate. Use the values with matching units.pressure = 600 ÷ 0.5 = 1200
- Write the answer with its unit. The unit is N/m².pressure = 1200 N/m²
Check. Work backwards: 1200 × 0.5 = 600 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the pressure (25 N/cm²) and the area (12 cm²).
We want the force. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.force = pressure × area
- Put the numbers in and calculate. Use the values with matching units.force = 25 × 12 = 300
- Write the answer with its unit. The unit is N.force = 300 N
Check. Work backwards: 300 ÷ 12 = 25 (the pressure). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Method. Change the mass unit and the volume unit separately.
- Change 8.9 g to kg. Grams to kg: divide by 1000.8.9 g = 0.0089 kg
- Change 1 cm³ to m³. 1 m³ = 1 000 000 cm³, so 1 cm³ is one millionth of a m³.1 cm³ = 0.000 001 m³
- Divide mass by volume. 0.0089 kg in 0.000 001 m³ means 1 000 000 times as many per m³.0.0089 × 1 000 000 = 8900 kg/m³
Check. The shortcut 1 g/cm³ = 1000 kg/m³ gives 8.9 × 1000 = 8900. ✓
Watch out. Remember the shortcut: g/cm³ × 1000 = kg/m³. Water is 1 g/cm³ = 1000 kg/m³.
Watch out
Check which quantity you are finding before you divide. Density is mass divided by volume, not volume divided by mass. If your density for a solid comes out as 0.02 or 500, check you have the right way round and the units match.Exercise 16G Density and pressure
Calculator allowedKey words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (120 g) and the volume (40 cm³).
We want the density. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.density = mass ÷ volume
- Put the numbers in and calculate. Use the values with matching units.density = 120 ÷ 40 = 3
- Write the answer with its unit. The unit is g/cm³.density = 3 g/cm³
Check. Work backwards: 3 × 40 = 120 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (780 g) and the volume (100 cm³).
We want the density. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.density = mass ÷ volume
- Put the numbers in and calculate. Use the values with matching units.density = 780 ÷ 100 = 7.8
- Write the answer with its unit. The unit is g/cm³.density = 7.8 g/cm³
Check. Work backwards: 7.8 × 100 = 780 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (90 g) and the volume (120 cm³).
We want the density. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.density = mass ÷ volume
- Put the numbers in and calculate. Use the values with matching units.density = 90 ÷ 120 = 0.75
- Write the answer with its unit. The unit is g/cm³.density = 0.75 g/cm³
Check. Work backwards: 0.75 × 120 = 90 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (2700 g) and the volume (1000 cm³).
We want the density. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.density = mass ÷ volume
- Put the numbers in and calculate. Use the values with matching units.density = 2700 ÷ 1000 = 2.7
- Write the answer with its unit. The unit is g/cm³.density = 2.7 g/cm³
Check. Work backwards: 2.7 × 1000 = 2700 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the density (8 g/cm³) and the volume (25 cm³).
We want the mass. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.mass = density × volume
- Put the numbers in and calculate. Use the values with matching units.mass = 8 × 25 = 200
- Write the answer with its unit. The unit is g.mass = 200 g
Check. Work backwards: 200 ÷ 25 = 8 (the density). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the density (2.5 g/cm³) and the volume (60 cm³).
We want the mass. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.mass = density × volume
- Put the numbers in and calculate. Use the values with matching units.mass = 2.5 × 60 = 150
- Write the answer with its unit. The unit is g.mass = 150 g
Check. Work backwards: 150 ÷ 60 = 2.5 (the density). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the density (0.9 g/cm³) and the volume (200 cm³).
We want the mass. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.mass = density × volume
- Put the numbers in and calculate. Use the values with matching units.mass = 0.9 × 200 = 180
- Write the answer with its unit. The unit is g.mass = 180 g
Check. Work backwards: 180 ÷ 200 = 0.9 (the density). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the density (19.3 g/cm³) and the volume (10 cm³).
We want the mass. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.mass = density × volume
- Put the numbers in and calculate. Use the values with matching units.mass = 19.3 × 10 = 193
- Write the answer with its unit. The unit is g.mass = 193 g
Check. Work backwards: 193 ÷ 10 = 19.3 (the density). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (360 g) and the density (9 g/cm³).
We want the volume. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.volume = mass ÷ density
- Put the numbers in and calculate. Use the values with matching units.volume = 360 ÷ 9 = 40
- Write the answer with its unit. The unit is cm³.volume = 40 cm³
Check. Work backwards: 9 × 40 = 360 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (56 g) and the density (0.8 g/cm³).
We want the volume. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.volume = mass ÷ density
- Put the numbers in and calculate. Use the values with matching units.volume = 56 ÷ 0.8 = 70
- Write the answer with its unit. The unit is cm³.volume = 70 cm³
Check. Work backwards: 0.8 × 70 = 56 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (5400 g) and the density (2.7 g/cm³).
We want the volume. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.volume = mass ÷ density
- Put the numbers in and calculate. Use the values with matching units.volume = 5400 ÷ 2.7 = 2000
- Write the answer with its unit. The unit is cm³.volume = 2000 cm³
Check. Work backwards: 2.7 × 2000 = 5400 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Density tells you how much mass there is in each unit of volume. A material with a high density, like iron, is heavy for its size. Mass is the amount of matter (in g or kg); volume is the space it takes up.
Method. Use the formula triangle for mass, density and volume: mass = density × volume. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the mass (193 g) and the density (19.3 g/cm³).
We want the volume. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.volume = mass ÷ density
- Put the numbers in and calculate. Use the values with matching units.volume = 193 ÷ 19.3 = 10
- Write the answer with its unit. The unit is cm³.volume = 10 cm³
Check. Work backwards: 19.3 × 10 = 193 (the mass). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (500 N) and the area (2 m²).
We want the pressure. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.pressure = force ÷ area
- Put the numbers in and calculate. Use the values with matching units.pressure = 500 ÷ 2 = 250
- Write the answer with its unit. The unit is N/m².pressure = 250 N/m²
Check. Work backwards: 250 × 2 = 500 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (90 N) and the area (3 cm²).
We want the pressure. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.pressure = force ÷ area
- Put the numbers in and calculate. Use the values with matching units.pressure = 90 ÷ 3 = 30
- Write the answer with its unit. The unit is N/cm².pressure = 30 N/cm²
Check. Work backwards: 30 × 3 = 90 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (1200 N) and the area (0.4 m²).
We want the pressure. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.pressure = force ÷ area
- Put the numbers in and calculate. Use the values with matching units.pressure = 1200 ÷ 0.4 = 3000
- Write the answer with its unit. The unit is N/m².pressure = 3000 N/m²
Check. Work backwards: 3000 × 0.4 = 1200 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (48 N) and the area (16 cm²).
We want the pressure. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.pressure = force ÷ area
- Put the numbers in and calculate. Use the values with matching units.pressure = 48 ÷ 16 = 3
- Write the answer with its unit. The unit is N/cm².pressure = 3 N/cm²
Check. Work backwards: 3 × 16 = 48 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the pressure (15 N/cm²) and the area (6 cm²).
We want the force. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.force = pressure × area
- Put the numbers in and calculate. Use the values with matching units.force = 15 × 6 = 90
- Write the answer with its unit. The unit is N.force = 90 N
Check. Work backwards: 90 ÷ 6 = 15 (the pressure). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the pressure (200 N/m²) and the area (3.5 m²).
We want the force. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.force = pressure × area
- Put the numbers in and calculate. Use the values with matching units.force = 200 × 3.5 = 700
- Write the answer with its unit. The unit is N.force = 700 N
Check. Work backwards: 700 ÷ 3.5 = 200 (the pressure). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the pressure (8 N/cm²) and the area (25 cm²).
We want the force. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.force = pressure × area
- Put the numbers in and calculate. Use the values with matching units.force = 8 × 25 = 200
- Write the answer with its unit. The unit is N.force = 200 N
Check. Work backwards: 200 ÷ 25 = 8 (the pressure). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (600 N) and the pressure (150 N/m²).
We want the area. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.area = force ÷ pressure
- Put the numbers in and calculate. Use the values with matching units.area = 600 ÷ 150 = 4
- Write the answer with its unit. The unit is m².area = 4 m²
Check. Work backwards: 150 × 4 = 600 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (72 N) and the pressure (6 N/cm²).
We want the area. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.area = force ÷ pressure
- Put the numbers in and calculate. Use the values with matching units.area = 72 ÷ 6 = 12
- Write the answer with its unit. The unit is cm².area = 12 cm²
Check. Work backwards: 6 × 12 = 72 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Key words. Force is a push or pull, measured in newtons (N). Pressure tells you how much force is acting on each unit of area. The same force on a small area gives a large pressure.
Method. Use the formula triangle for force, pressure and area: force = pressure × area. Cover what you want to find to see the calculation.
- Write down what you know and what you want. Always list the quantities with their units before you start.We know the force (2000 N) and the pressure (500 N/m²).
We want the area. - Choose the formula. Cover the quantity you want in the formula triangle. What is left tells you whether to multiply or divide.area = force ÷ pressure
- Put the numbers in and calculate. Use the values with matching units.area = 2000 ÷ 500 = 4
- Write the answer with its unit. The unit is m².area = 4 m²
Check. Work backwards: 500 × 4 = 2000 (the force). ✓
Watch out. Check the units match before you calculate (for example, minutes must be changed to hours if the speed is per hour).
Method. Find the volume of the cuboid first, then divide the mass by the volume.
- Find the volume. Volume of a cuboid = length × width × height.5 × 4 × 3 = 60 cm³
- Use density = mass ÷ volume. The mass is 468 g.468 ÷ 60 = 7.8
- Write the answer with units. Mass in g and volume in cm³ give g/cm³.7.8 g/cm³
Check. 7.8 × 60 = 468 g. ✓
Method. Use mass = density × volume, then change grams to kilograms.
- Calculate the mass in grams. Mass = density × volume.19.3 × 250 = 4825 g
- Change to kg. Grams to kilograms: divide by 1000.4825 ÷ 1000 = 4.825 kg
Check. Rough check: 20 × 250 = 5000 g = 5 kg, close to 4.825 kg. ✓
Method. Work out the density of the block and compare it with the density of water.
- Find the density. Density = mass ÷ volume.450 ÷ 500 = 0.9 g/cm³
- Compare with water. Less dense than water floats; more dense sinks.0.9 < 1
- Write the conclusion. The block floats.
Check. 500 cm³ of water would have a mass of 500 g, which is more than the 450 g block, so the block is lighter than the water it pushes aside. ✓
Method. Work out both densities and compare.
- Density of A. Density = mass ÷ volume.240 ÷ 80 = 3 g/cm³
- Density of B. Same formula.150 ÷ 60 = 2.5 g/cm³
- Compare. 3 > 2.5, so A is denser.
Check. If both were 600 cm³, A would have mass 1800 g and B 1500 g. ✓
Method. Total area is the area of both feet.
- Find the total area. Two feet touch the ground.2 × 200 = 400 cm²
- Pressure = force ÷ area. 500 ÷ 400 = 1.25 N/cm²
Check. 1.25 × 400 = 500 N. ✓
Method. Only one foot touches the ground, so the area is halved.
- Use the area of one foot. 200 cm²
- Pressure = force ÷ area. 500 ÷ 200 = 2.5 N/cm²
Check. Half the area gives double the pressure: 2 × 1.25 = 2.5. ✓
Method. Work out the pressure for each shoe, then compare.
- Flat shoes. Pressure = force ÷ area.600 ÷ 100 = 6 N/cm²
- Stiletto heels. Same force, much smaller area.600 ÷ 1 = 600 N/cm²
- Compare and explain. The same force is concentrated onto an area 100 times smaller, so the pressure is 100 times greater and can dent the floor.600 ÷ 6 = 100
Check. 100 times smaller area means 100 times larger pressure. ✓
Method. Find the volume in m³, use mass = density × volume, then change kg to tonnes.
- Find the volume. Length × width × depth.10 × 5 × 2 = 100 m³
- Find the mass. Mass = density × volume, in kg because the density is in kg/m³.1000 × 100 = 100000 kg
- Change to tonnes. 1 tonne = 1000 kg, so divide by 1000.100 000 ÷ 1000 = 100 tonnes
Check. 1 m³ of water is 1 tonne, so 100 m³ is 100 tonnes. ✓
Method. 1 g/cm³ = 1000 kg/m³, so multiply by 1000.
- Change the mass. 1 g = 0.001 kg.8.9 g = 0.0089 kg
- Change the volume. 1 cm³ = 0.000 001 m³.0.0089 ÷ 0.000 001 = 8900
- Write the answer. 8900 kg/m³
Check. Water: 1 g/cm³ = 1000 kg/m³ matches the shortcut. ✓
Method. Find the mass of each liquid, add the masses and the volumes, then divide.
- Mass of A. Mass = density × volume.0.8 × 200 = 160 g
- Mass of B. Same formula.1.2 × 300 = 360 g
- Totals. Add the masses and add the volumes.mass = 160 + 360 = 520 g
volume = 200 + 300 = 500 cm³ - Density of the mixture. Density = total mass ÷ total volume.520 ÷ 500 = 1.04 g/cm³
Check. The answer lies between 0.8 and 1.2 and is closer to 1.2 because there is more of B. ✓
Watch out. Do not average the two densities. Find the total mass and total volume.
Method. There are 10 000 cm² in 1 m² (100 × 100), so 1 m² holds 10 000 squares of 1 cm², each with 3 N on it.
- Write the area fact. 1 m = 100 cm so 1 m² = 100 × 100 cm².1 m² = 10 000 cm²
- Multiply. Each cm² carries 3 N and there are 10 000 of them in 1 m².3 × 10 000 = 30000 N/m²
Check. 1 N/cm² = 10 000 N/m², so 3 N/cm² = 30 000 N/m². ✓
Watch out. Pressure per cm² becomes a bigger number per m²: multiply by 10 000, not divide.