Chapter 5 of 27

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ICSE Class 8 Mathematics★ ICSE Class 8 Mathematics · Full Year Coverage · Board: ICSE

Geometry, Mensuration, Data Handling & Probability

The geometry of QUADRILATERALS — parallelograms, rhombuses, rectangles, squares. Building 3D shapes. Measuring the SURFACE AREA and VOLUME of solids. And an introduction to PROBABILITY. ICSE Class 8 Mathematics.

⏱ 16 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1State properties of parallelogram, rhombus, rectangle, square, trapezium, and kite
2Apply the angle sum formula for polygons: (n-2)×180°
3Calculate CSA, TSA, and Volume for cuboid, cube, and cylinder
4Apply divisibility tests for 2, 3, 4, 5, 9, 10, and 11
5Organise data using tally/frequency tables; read histograms and pie charts
6Calculate mean, median, and mode from grouped and ungrouped data
7Define probability; calculate P(E) = favourable outcomes/total outcomes for standard experiments

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Why this chapter matters

This ICSE Class 8 Mathematics chapter covers quadrilateral properties, 3D mensuration, data handling, and probability — all high-scoring topics in the annual exam. The quadrilateral properties table (parallelogram/rhombus/rectangle/square) is tested every year. Cylinder surface area and volume (CSA=2πrh, TSA=2πr(r+h), V=πr²h) is a reliable 3-4 mark calculation. Probability basics (P(E) = favourable/total outcomes) with dice and coin problems appear in every exam. Divisibility tests (especially by 3, 9, 11) are MCQ staples. The general formula for polygon angle sum ((n-2)×180°) extends from triangles to all polygons.

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Geometry, Mensuration & Data Handling

1. Understanding Quadrilaterals

Properties

Shape	Properties
Parallelogram	Opposite sides ∥ and =. Opposite ∠s =. Diagonals BISECT each other.
Rhombus	Parallelogram + ALL sides =. Diagonals PERPENDICULAR. Diagonals BISECT angles.
Rectangle	Parallelogram + ALL angles = 90°. Diagonals =.
Square	Rectangle + Rhombus. All sides =. All ∠s = 90°. Diagonals =, ⟂, bisect ∠s.
Trapezium	ONE pair of parallel sides.
Kite	TWO PAIRS of adjacent EQUAL sides. One diagonal bisects the other.

Angle Sum

Quadrilateral: 360°. Pentagon: (5-2)×180° = 540°. n-gon: (n-2)×180°.

2. Practical Geometry

Constructing Quadrilaterals

When given: (a) 4 sides + 1 diagonal. (b) 3 sides + 2 diagonals. (c) 3 sides + 2 included angles. (d) 2 adjacent sides + 3 angles.

Use: RULER, COMPASS, PROTRACTOR. 'Construction is a step-by-step process. Draw the base first. Then build outward. Every measurement must be EXACT.'

3. Mensuration — Surface Area and Volume

Cuboid (Box)

Surface Area = 2(lb + bh + hl). Volume = l × b × h.
Diagonal = √(l² + b² + h²).

Cube

Surface Area = 6s². Volume = s³.

Cylinder

Curved Surface Area (CSA) = 2πrh
Total Surface Area (TSA) = 2πr(r + h)
Volume = πr²h

Units

Measure	Units
Area	cm², m², km²
Volume	cm³, m³, litres (1 m³ = 1,000 L)

4. Playing with Numbers

Tests of Divisibility

By	Condition
2	Last digit even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Last digit 0 or 5
9	Sum of digits divisible by 9
10	Last digit 0
11	Difference between sum of digits at odd and even places = 0 or multiple of 11

Number Puzzles

Numbers can be written in GENERALISED FORM: ab = 10a + b. 'Number puzzles and games strengthen number sense.'

5. Data Handling

Organising and Grouping Data

Array. Tally marks. Frequency distribution.
Class intervals (grouped data).

Graphical Representation

Bar graph: COMPARING categories
Histogram: Continuous data. NO gap between bars.
Pie chart: Showing PROPORTIONS of a whole.

Measures of Central Tendency

Mean = Sum / Number
Median = Middle value (when ordered)
Mode = Most frequent value

6. Introduction to Probability

What Is Probability?

A measure of how LIKELY an event is to occur. P(E) = Number of favourable outcomes / Total number of outcomes.

'Probability is a number between 0 and 1.'

Random Experiments

Tossing a coin: P(Head) = 1/2. P(Tail) = 1/2.
Rolling a die: P(6) = 1/6. P(even) = 3/6 = 1/2.
Drawing a card from a standard 52-card deck.

Impossible and Certain Events

Impossible: P = 0 (e.g., getting 7 on a standard die)
Certain: P = 1 (e.g., the sun rising tomorrow)
'All other events have probability BETWEEN 0 and 1.'

Key formulas & results

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Quadrilaterals, 3D Mensuration, Data, and Probability

QUADRILATERAL PROPERTIES: PARALLELOGRAM: opp sides ∥ and equal; opp angles equal; consecutive angles supplementary (180°); diagonals BISECT each other. RHOMBUS: all properties of parallelogram + ALL SIDES equal + diagonals PERPENDICULAR (⊥) + diagonals BISECT the angles. RECTANGLE: all properties of parallelogram + ALL ANGLES = 90° + diagonals EQUAL in length. SQUARE: all properties of rectangle AND rhombus (all sides equal, all angles 90°, diagonals equal AND perpendicular AND bisect angles). TRAPEZIUM: ONE pair of parallel sides (unlike parallelogram which has TWO pairs). KITE: TWO PAIRS of ADJACENT equal sides; one diagonal bisects the other at right angles. POLYGON ANGLE SUM: (n-2) × 180° where n = number of sides. Triangle (n=3): 180°. Quadrilateral (n=4): 360°. Pentagon (n=5): 540°. Hexagon (n=6): 720°. MENSURATION — 3D SHAPES: CUBOID: SA = 2(lb + bh + hl). V = l×b×h. CUBE: SA = 6s². V = s³. CYLINDER: CSA (curved only) = 2πrh. TSA = 2πr(r+h) = 2πrh + 2πr². V = πr²h. UNITS: Area in cm²/m². Volume in cm³/m³/litres. 1m³ = 1,000 L. DIVISIBILITY TESTS: By 2: last digit even. By 3: sum of digits divisible by 3. By 4: last two digits divisible by 4. By 5: last digit 0 or 5. By 9: sum of digits divisible by 9. By 10: last digit 0. By 11: |sum of odd-position digits − sum of even-position digits| = 0 or multiple of 11. DATA HANDLING: Bar graph (comparing categories). Histogram (continuous data — NO gap between bars). Pie chart (proportions of a whole). MEAN = sum/count. MEDIAN = middle value when ordered. MODE = most frequent. PROBABILITY: P(E) = Number of FAVOURABLE outcomes / Total number of possible outcomes. Range: 0 ≤ P(E) ≤ 1. P = 0: IMPOSSIBLE event. P = 1: CERTAIN event. COIN: P(H) = P(T) = 1/2. DIE: P(any one number) = 1/6. P(even) = 3/6 = 1/2. P(prime) = P(2,3,5) = 3/6 = 1/2. DECK OF CARDS (52): P(a specific card) = 1/52. P(any ace) = 4/52 = 1/13. P(heart) = 13/52 = 1/4.

ICSE CLASS 8 MATHS KEY TRAPS: (1) RHOMBUS diagonals are PERPENDICULAR but NOT necessarily equal. RECTANGLE diagonals are EQUAL but NOT perpendicular. SQUARE: both equal AND perpendicular. (2) HISTOGRAM vs BAR GRAPH: Histogram = continuous data, NO GAPS between bars. Bar graph = discrete/categorical data, GAPS between bars. (3) CYLINDER TSA = 2πr(r+h) includes BOTH circular bases. If the question asks for CSA (just the curved part), use 2πrh only. (4) DIVISIBILITY BY 11: It's |alternate sum|. E.g., 29645: (9+4) − (2+6+5) = 13 − 13 = 0 → divisible by 11 ✓. (5) PROBABILITY: favourable outcomes ≤ total outcomes always. Never > 1.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Saying a rhombus has equal diagonals, or that a rectangle's diagonals are perpendicular

✓ DIAGONAL PROPERTIES of the special quadrilaterals: RECTANGLE: Diagonals are EQUAL in length but NOT perpendicular. They bisect each other (like all parallelograms). RHOMBUS: Diagonals are PERPENDICULAR (meet at 90°) but NOT equal in length. They bisect each other AND they bisect the vertex angles. SQUARE: Diagonals are BOTH equal AND perpendicular. They bisect each other and the vertex angles. Memory aid: RECTANGLE = R for 'Right angles' (all 4 angles are 90°) but the diagonals are NOT right angles. RHOMBUS = all sides equal, the diagonals 'cut right across' each other at 90°. The SQUARE inherits BOTH properties.

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1MEDIUM· cylinder-probability

A cylindrical water tank has a radius of 3.5 m and a height of 10 m. Find: (a) the curved surface area, (b) the total surface area, (c) the volume in m³ and litres. (Use π = 22/7). Also: A die is rolled once. Find the probability that the number is: (d) a multiple of 3, (e) greater than 4.

▶ Show solution

CYLINDER: r = 3.5 m, h = 10 m, π = 22/7. (a) CSA = 2πrh = 2 × (22/7) × 3.5 × 10 = 2 × (22/7) × 35 = 2 × 110 = 220 m². (b) TSA = 2πr(r+h) = 2 × (22/7) × 3.5 × (3.5+10) = 2 × (22/7) × 3.5 × 13.5 = 2 × (22/7) × 47.25 = 2 × 148.5 = 297 m². [OR: TSA = CSA + 2πr² = 220 + 2 × (22/7) × 3.5² = 220 + 2 × (22/7) × 12.25 = 220 + 77 = 297 m² ✓] (c) VOLUME = πr²h = (22/7) × 3.5² × 10 = (22/7) × 12.25 × 10 = (22/7) × 122.5 = 385 m³. In litres: 385 m³ × 1000 = 385,000 litres. PROBABILITY (die): Total outcomes = 6 (numbers 1,2,3,4,5,6). (d) Multiples of 3: 3 and 6 → 2 favourable outcomes. P(multiple of 3) = 2/6 = 1/3. (e) Greater than 4: 5 and 6 → 2 favourable outcomes. P(>4) = 2/6 = 1/3.

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