Mensuration and trigonometry Revision Notes

What are Units of Measurement?

Units of measurement are standard quantities used to express the magnitude of physical properties. They give context and meaning to numbers. For example, saying '5 kg' unambiguously refers to a mass, whereas just '5' is unclear. They are essential for accurate communication and calculation in mathematics and science. Understanding the base units for common measurements is the first step.

Common Units and Conversions (Metric System)

The metric system is widely used and relies on powers of 10 for conversions. Key units to know:

• Length: millimetres (mm), centimetres (cm), metres (m), kilometres (km). Conversions: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m.
• Mass: grams (g), kilograms (kg), tonnes (t). Conversions: 1 kg = 1000 g, 1 t = 1000 kg.
• Capacity/Volume: millilitres (ml), litres (l), cubic centimetres (cm³), cubic metres (m³). Conversions: 1 l = 1000 ml, 1 l = 1000 cm³, 1 m³ = 1000 l.
• Time: seconds (s), minutes (min), hours (h), days, weeks, months, years. Conversions: 1 min = 60 s, 1 h = 60 min, 1 day = 24 h.

Mastering these conversions is crucial.

Applying Units in Calculations

When performing calculations (like adding, subtracting, multiplying, or dividing), all quantities involved must be expressed in the same units. If they are not, the first step is to convert them to a common unit. For example, to add 0.5 m and 30 cm, you should convert 0.5 m to 50 cm, then add 50 cm + 30 cm = 80 cm. Always include the correct unit in your final answer to ensure clarity and accuracy. Incorrect or missing units can lead to errors and loss of marks.

Reasoning with Units

Reasoning with units involves selecting the most appropriate unit for a given context and justifying your choice. For example, it makes sense to measure the distance between cities in kilometres, not millimetres, because kilometres provide a more manageable and understandable number. Similarly, a spoonful of medicine would be measured in millilitres, not litres. You should be able to explain why a particular unit is suitable for a specific measurement and why other units might be less appropriate.

Common Metric Units

The metric system is a decimal-based system used widely. For length, remember: 1 kilometre (km) = 1000 metres (m), 1 metre (m) = 100 centimetres (cm), 1 centimetre (cm) = 10 millimetres (mm). For mass: 1 kilogram (kg) = 1000 grams (g). For capacity: 1 litre (L) = 1000 millilitres (ml). Conversions within the metric system simply involve multiplying or dividing by powers of 10. For example, to convert 500 cm to metres, divide by 100 (500 ÷ 100 = 5 m).

Common Imperial Units

The imperial system is still used for some measurements, especially in certain contexts. For length: 1 foot (ft) = 12 inches (in), 1 yard (yd) = 3 feet, 1 mile (mi) = 1760 yards. For mass: 1 pound (lb) = 16 ounces (oz), 1 stone (st) = 14 pounds. For capacity: 1 gallon (gal) = 8 pints (pt). Unlike the metric system, these conversions do not follow a simple base-10 rule and require memorisation or reference to specific conversion factors.

Metric to Imperial Conversions

You need to be able to convert between metric and imperial units using approximate conversion factors. Key conversions to remember or know how to use include: Length: 1 inch ≈ 2.5 cm, 1 mile ≈ 1.6 km. Mass: 1 kg ≈ 2.2 pounds. Capacity: 1 litre ≈ 1.76 pints. When performing conversions, make sure to consider whether you need to multiply or divide. For example, to convert 10 inches to cm, multiply by 2.5 (10 * 2.5 = 25 cm). To convert 88 pounds to kg, divide by 2.2 (88 / 2.2 = 40 kg).

Choosing Appropriate Units

Selecting the correct unit for a measurement is an important part of understanding context. For large distances, like between cities, kilometres or miles are appropriate. For the height of a person or a room, metres or feet are suitable. For small items like a coin, millimetres or centimetres would be used. Similarly, for the mass of a person, kilograms or stone are appropriate, while for a very light object, grams or ounces are better. For liquid volumes, litres or pints are used for larger quantities, and millilitres for smaller ones. Always think about the scale of the object or quantity being measured.

What are Compound Units?

Compound units combine two or more basic units to describe a rate or relationship between different physical quantities. Examples include speed (distance/time), density (mass/volume), and pressure (force/area). Understanding the components of these units helps in setting up calculations correctly. For example, if speed is in km/h, it tells you distance in kilometres divided by time in hours. The '/' symbol means 'per' and signifies division.

Calculating Speed, Distance, and Time

Speed is defined as distance travelled per unit time. The formula is Speed = Distance / Time. This can be rearranged to find distance (Distance = Speed × Time) or time (Time = Distance / Speed). It is crucial to ensure that units are consistent. If speed is given in m/s, distance should be in metres and time in seconds. If speed is in km/h, distance should be in kilometres and time in hours. Converting units before or after calculation is often necessary.

Calculating Density, Mass, and Volume

Density is a measure of how much mass is contained per unit volume. The formula is Density = Mass / Volume. This can be rearranged to find mass (Mass = Density × Volume) or volume (Volume = Mass / Density). Common units for density include g/cm³ or kg/m³. Similar to speed calculations, keeping units consistent is vital. If density is in g/cm³, then mass should be in grams and volume in cubic centimetres.

Applying Compound Unit Reasoning

When solving problems involving compound units, identify the quantities given and the quantity to be calculated. Write down the relevant formula. Check if any unit conversions are needed to ensure consistency (e.g., minutes to hours, cm to m). Perform the calculation and state the answer with the correct compound unit. Always think about whether your answer makes sense in the context of the problem.

What are Compound Units?

Compound units combine two or more basic units to describe a rate or relationship between different quantities. The word 'per' often indicates division. For example, speed is 'distance per time', meaning distance divided by time. This is represented by units like metres per second (m/s) or miles per hour (mph). Another common compound unit is density, which is 'mass per volume', usually measured in grams per cubic centimetre (g/cm3) or kilograms per cubic metre (kg/m3). Understanding the relationship between the quantities is crucial.

Calculating Speed, Distance, and Time

Speed, distance, and time are related by the formula: Speed = Distance / Time. This can be rearranged to find any of the three quantities.

• To find Distance: Distance = Speed x Time
• To find Time: Time = Distance / Speed

Always ensure that your units are consistent. For example, if speed is in km/h and time is in hours, your distance will be in km.

Calculating Density, Mass, and Volume

Density, mass, and volume are also related through a compound unit. The formula is: Density = Mass / Volume. This formula can also be rearranged:

• To find Mass: Mass = Density x Volume
• To find Volume: Volume = Mass / Density

When performing calculations, make sure your units align. If density is in g/cm3 and volume is in cm3, your mass will be in grams.

Other Compound Units (e.g. Pressure)

Pressure is another example of a compound unit, defined as Force per unit Area. This is represented by the formula: Pressure = Force / Area. The standard unit for pressure is the Pascal (Pa), which is equivalent to Newtons per square metre (N/m2). Similarly, if you know the pressure and the area, you can find the force: Force = Pressure x Area. Or, if you know the force and pressure, you can find the area: Area = Force / Pressure. These relationships are essential for problem-solving involving compound units.