Geometry, measurement and data handling

What this heading covers

Geometry and measurement are where a paper can be answered quickly if the formulas are secure and lost slowly if they are not. Almost nothing here needs derivation; it needs exactness.

The building blocks

• A point has position and no size. A line extends without end in both directions; a line segment has two endpoints; a ray has one endpoint and goes on in one direction.
• Parallel lines never meet. Intersecting lines meet at a point. Perpendicular lines meet at 90°.
• A plane is a flat surface extending in all directions.

Angles

Angle	Measure
Acute	less than 90°
Right	exactly 90°
Obtuse	between 90° and 180°
Straight	exactly 180°
Reflex	between 180° and 360°
Complete	360°

Complementary angles add to 90°; supplementary angles add to 180°. Swapping those two is one of the most frequent avoidable errors in the paper.

Also asked: vertically opposite angles (formed by two intersecting lines) are equal; adjacent angles share a vertex and an arm; linear pair angles are adjacent and supplementary.

Triangles

By sides: equilateral (all three equal, all angles 60°), isosceles (two equal), scalene (none equal).

By angles: acute (all under 90°), right (one 90°), obtuse (one over 90°).

Rules to hold exactly:

• Angle sum = 180°.
• An exterior angle equals the sum of the two opposite interior angles.
• Triangle inequality: the sum of any two sides is greater than the third. A question giving three lengths and asking whether a triangle is possible is applying this.
• Pythagoras, for a right triangle: hypotenuse² = base² + height². The common triples — 3-4-5, 5-12-13, 8-15-17 — are worth recognising on sight.
• Area = ½ × base × height. For two sides and the included angle, ½ × a × b × sin C.

Quadrilaterals

Angle sum = 360° for every quadrilateral.

Shape	Defining properties
Square	4 equal sides, 4 right angles, diagonals equal and bisect at 90°
Rectangle	Opposite sides equal, 4 right angles, diagonals equal
Parallelogram	Opposite sides parallel and equal, opposite angles equal, diagonals bisect each other
Rhombus	4 equal sides, opposite angles equal, diagonals bisect at 90°
Trapezium	Exactly one pair of parallel sides
Kite	Two pairs of adjacent sides equal

Every square is a rectangle and a rhombus; not every rectangle is a square. Questions are often built on that one-way relationship.

Circles

• Radius joins the centre to the circle; diameter = 2 × radius and passes through the centre.
• A chord joins two points on the circle; the diameter is the longest chord.
• An arc is part of the circle; a sector is bounded by two radii and an arc; a segment by a chord and an arc.
• Circumference = 2πr = πd. Area = πr². Take π as 22/7 or 3.14 as the question suggests.

Solids

Faces, edges and vertices are asked directly:

Solid	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Cylinder	3 (2 flat, 1 curved)	2	0
Cone	2 (1 flat, 1 curved)	1	1 (apex)
Sphere	1 (curved)	0	0
Square pyramid	5	8	5
Triangular prism	5	9	6

For any polyhedron, Euler's formula: F + V − E = 2. Check the cube: 6 + 8 − 12 = 2.

A net is the flat shape that folds into a solid; questions show a net and ask which solid it makes.

Perimeter, area and volume

Shape	Perimeter / circumference	Area
Square	4 × side	side²
Rectangle	2 × (length + breadth)	length × breadth
Triangle	sum of the three sides	½ × base × height
Circle	2πr	πr²
Parallelogram	2 × (a + b)	base × height
Trapezium	sum of sides	½ × (sum of parallel sides) × height

Volume: cube = side³; cuboid = l × b × h; cylinder = πr²h. Surface area of a cuboid = 2(lb + bh + hl); of a cube = 6 × side².

The distinction the paper probes: perimeter is a length, area is a length squared, volume is a length cubed — so their units are cm, cm² and cm³. Two shapes can share a perimeter and differ in area, and a question often sets exactly that.

Symmetry, reflection and construction

• A figure has line symmetry if a line divides it into two mirror halves. Lines of symmetry: square 4, rectangle 2, equilateral triangle 3, isosceles triangle 1, rhombus 2, circle infinitely many, scalene triangle none, parallelogram none.
• Rotational symmetry: the figure looks the same after turning by less than a full turn. The order is how many times in one full turn — square 4, equilateral triangle 3, rectangle 2.
• A reflection reverses left and right; the image is the same distance behind the mirror line.
• Construction uses a straight edge, ruler, protractor and compasses: bisecting a line segment, bisecting an angle, and copying an angle are the standard ones.

Measurement

Metric, and each step is a factor of 10:

km  →  hm  →  dam  →  m  →  dm  →  cm  →  mm      (÷10 each step to the right)

1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm. For mass, 1 kg = 1000 g and 1 g = 1000 mg; for capacity, 1 L = 1000 mL. Note that 1 litre = 1000 cm³, which is how volume and capacity questions connect.

Time: 60 seconds to a minute, 60 minutes to an hour, 24 hours to a day. The 24-hour clock runs 00:00 to 23:59; 12 noon is 12:00 and 12 midnight is 00:00. A leap year has 366 days and occurs when the year is divisible by 4 — except century years, which must be divisible by 400, so 1900 was not a leap year and 2000 was.

Money: word problems on cost, profit and loss, and simple interest. Profit or loss is always calculated on the cost price: profit % = (profit ÷ cost price) × 100. Simple interest = (P × R × T) ÷ 100.

Patterns

Number patterns, shape patterns and the rule behind them. The question is always "what comes next" or "what is the rule", and the method is to look at the difference between consecutive terms, then at the difference of the differences if the first is not constant.

Worth knowing: the sum of the first n natural numbers is n(n + 1)/2, and consecutive odd numbers from 1 sum to perfect squares (1, 4, 9, 16).

Data handling

• Pictograph uses a symbol for a number of items; read the key before answering.
• Bar graph compares categories; the axis scale is where mistakes happen.
• Tally marks group in fives, the fifth struck across the other four.
• Pie chart shows parts of a whole; the whole circle is 360°, so a category's angle is its fraction of the total × 360.

The three averages:

• Mean = sum ÷ number of values.
• Median = the middle value after sorting; with an even count, the mean of the two middle values.
• Mode = the value occurring most often. A set can have more than one mode, or none.

Sorting first is the step candidates skip, and a median question is usually designed to punish exactly that.

A question of the kind the paper sets

The perimeter of a rectangle is 36 cm and its length is twice its breadth. What is its area?

Let the breadth be b, so the length is 2b. Perimeter = 2(l + b) = 2(2b + b) = 6b = 36, giving b = 6 cm and l = 12 cm. Area = 12 × 6 = 72 cm².

Two traps sit in this. Dividing 36 by 2 and calling it the length ignores that the perimeter counts each side twice. And the answer is an area, so the unit is cm² — an option offering "72 cm" is there for the candidate who stopped thinking one step early.

Mistakes that cost marks here

• Swapping complementary (90°) and supplementary (180°).
• Giving an area in cm or a volume in cm².
• Finding the median without sorting the values first.
• Forgetting that a square is also a rectangle and a rhombus.
• Counting a cylinder as having vertices, or a cone as having none.
• Treating every century year as a leap year.
• Calculating profit percentage on the selling price. It is always on the cost price.

In one line

This heading rewards exactness rather than cleverness: the formulas are few, the definitions are fixed, and most lost marks come from a swapped pair of terms or a forgotten unit.

For the classroom

Every formula here was once a discovery: perimeter by walking the edge of the classroom, area by covering a desk with square tiles, volume by filling a box with cubes. The pedagogy paper asks why that sequence matters; this one asks for the formula it ends in.