Geometry, Measurement & Data

Measurement & Data

Grade 7 · Grade 8 · Grade 9

✓Calculate the area, surface area, and volume of various 2D and 3D shapes.
✓Collect, organise, represent, and interpret data using various graphical methods.
✓Determine measures of central tendency (mean, median, mode) and range for a given data set.
✓Understand and calculate the theoretical and experimental probability of simple events.
✓Apply appropriate units of measurement in calculations and answers.

Key concepts

Area

Area is the amount of two-dimensional space a flat surface or shape covers. It is measured in square units (e.g., cm², m², km²).

Square: Area = side × side = s²\nRectangle: Area = length × breadth = l × b\nTriangle: Area = ½ × base × perpendicular height = ½bh\nCircle: Area = πr² (where π ≈ 3.14 or 22/7, and r is the radius)

Surface Area

Surface area is the total area of all the faces (surfaces) of a three-dimensional object. It is also measured in square units (e.g., cm², m²).

Cube: SA = 6s² (where s is the side length)\nRectangular Prism: SA = 2(lb + lh + bh) (where l=length, b=breadth, h=height)\nCylinder: SA = 2πr² + 2πrh (where r=radius, h=height)

Volume

Volume is the amount of three-dimensional space an object occupies. It is measured in cubic units (e.g., cm³, m³, km³).

Cube: Volume = s³\nRectangular Prism: Volume = l × b × h\nCylinder: Volume = πr²h

Data Handling

Data handling involves collecting, organising, representing, analysing, and interpreting information (data). Key terms include: Data (facts or information collected), Tally Table (records frequencies using tally marks), Frequency Table (shows how often each value appears), Bar Graph (uses bars for comparisons), Pie Chart (shows parts of a whole), Line Graph (shows trends over time), Histogram (for continuous data, no gaps between bars), Stem-and-Leaf Plot (shows distribution of numerical data).

Measures of Central Tendency

These are single values that describe the centre of a data set.

Mean = (Sum of all values) / (Number of values)\nMedian: The middle value in an ordered data set. If an even number of values, it is the average of the two middle values.\nMode: The value that appears most frequently in a data set.

Range

The difference between the highest and lowest values in a data set, indicating the spread of the data.

Range = Highest value - Lowest value

Probability

Probability is the measure of the likelihood that an event will occur. It is expressed as a fraction, decimal, or percentage between 0 (impossible) and 1 (certain). Key terms include: Outcome (a possible result), Event (a specific outcome or set of outcomes), Sample Space (the set of all possible outcomes).

Theoretical Probability: P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes)\nExperimental Probability (Relative Frequency): P(Event) = (Number of times the event occurred) / (Total number of trials)

Key facts to remember

1Area is measured in square units (e.g., cm², m²), while volume is measured in cubic units (e.g., cm³, m³).
2Always include appropriate units in your final answers for measurements.
3To find the median, data must first be ordered from smallest to largest.
4The mode is the value that appears most often; a data set can have no mode or multiple modes.
5Probability values always lie between 0 and 1, inclusive.
6π (pi) is a constant used in calculations involving circles and cylinders, often approximated as 3.14 or 22/7.
7The sum of probabilities of all possible outcomes in an experiment is always 1.
8A histogram has no gaps between its bars because it represents continuous data.

Worked examples

Example 1

A rectangular swimming pool is 10 m long, 5 m wide, and 1.5 m deep.\na) Calculate the area of the base of the pool.\nb) Calculate the volume of water needed to fill the pool.\nc) Calculate the surface area of the inside of the pool (base and four sides).

Ia) Area of base = length × breadth

II Area of base = 10 m × 5 m

III Area of base = 50 m²

IVb) Volume = length × breadth × height

V Volume = 10 m × 5 m × 1.5 m

VI Volume = 75 m³

VIIc) Surface Area of inside = Area of base + Area of 2 long sides + Area of 2 short sides

VIII Surface Area of inside = (10 × 5) + 2(10 × 1.5) + 2(5 × 1.5)

9 Surface Area of inside = 50 + 2(15) + 2(7.5)

10 Surface Area of inside = 50 + 30 + 15

11 Surface Area of inside = 95 m²

Answer

a) 50 m²\nb) 75 m³\nc) 95 m²

Example 2

The marks (out of 20) of 10 students in a Mathematics test are: 15, 12, 18, 10, 15, 19, 12, 15, 17, 13.\na) Determine the mean mark.\nb) Determine the median mark.\nc) Determine the mode mark.\nd) Determine the range of the marks.

IFirst, order the data: 10, 12, 12, 13, 15, 15, 15, 17, 18, 19.

IIa) Mean = (Sum of all values) / (Number of values)

III Mean = (10 + 12 + 12 + 13 + 15 + 15 + 15 + 17 + 18 + 19) / 10

IV Mean = 146 / 10

V Mean = 14.6

VIb) Median: There are 10 data points (an even number). The median is the average of the 5th and 6th values.

VII 5th value = 15

VIII 6th value = 15

9 Median = (15 + 15) / 2 = 15

10c) Mode: The value that appears most frequently.

11 The value 15 appears 3 times, which is more than any other value.

12 Mode = 15

13d) Range = Highest value - Lowest value

14 Range = 19 - 10

15 Range = 9

Answer

a) Mean = 14.6\nb) Median = 15\nc) Mode = 15\nd) Range = 9

Always order data first when finding the median or range.

Example 3

A bag contains 4 red, 3 blue, and 5 green marbles. A marble is chosen at random from the bag.\na) What is the total number of marbles in the bag?\nb) What is the probability of choosing a red marble?\nc) What is the probability of choosing a blue or green marble?

Ia) Total number of marbles = Number of red + Number of blue + Number of green

II Total number of marbles = 4 + 3 + 5 = 12 marbles.

IIIb) P(Red) = (Number of favourable outcomes) / (Total number of possible outcomes)

IV P(Red) = (Number of red marbles) / (Total number of marbles)

V P(Red) = 4 / 12

VI P(Red) = 1/3

VIIc) P(Blue or Green) = (Number of blue marbles + Number of green marbles) / (Total number of marbles)

VIII P(Blue or Green) = (3 + 5) / 12

9 P(Blue or Green) = 8 / 12

10 P(Blue or Green) = 2/3

Answer

a) 12 marbles\nb) 1/3\nc) 2/3

Probabilities should always be simplified to their lowest fraction form.

Common mistakes

✗Confusing area with perimeter, or surface area with volume.
✗Not ordering data before finding the median.
✗Forgetting to include units or using incorrect units in answers.
✗Incorrectly applying formulas, especially for composite shapes or 3D objects.
✗Calculating the mean instead of the median or mode, or vice-versa.
✗Expressing probability as a number greater than 1 or less than 0.

Exam tips

★Read questions carefully to identify whether you need to calculate area, surface area, or volume.
★Always show all your working steps clearly, as marks are often awarded for method.
★When working with data, organise it systematically (e.g., order data for median, create frequency tables).
★For probability questions, clearly state the number of favourable outcomes and the total number of possible outcomes.
★Use the correct formulas and substitute values accurately, paying attention to units.
★Double-check your calculations, especially for multi-step problems.

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