1. Measurement Units

Measurement is the process of assigning numbers to physical quantities. In mathematics, understanding measurement units is essential for accurate calculations and real-life problem-solving.

Key Standard Units in the Ghanaian Context:

Subtopics and Explanations

1. Length

   • Measures distance between two points.

   • Conversion: 1 m = 100 cm, 1 km = 1000 m

   • Example: A classroom is 8 m long and 6 m wide. Find its perimeter and area.

2. Mass

   • Measures how heavy an object is.

   • Conversion: 1 kg = 1000 g

   • Example: A bag of rice weighs 12.5 kg. How many grams is this? 12.5 × 1000 = 12,500 g

3. Capacity

   • Measures the volume of liquid an object can hold.

   • Conversion: 1 L = 1000 mL

   • Example: A jug holds 2.5 L of water. How many mL? 2.5 × 1000 = 2500 mL

4. Time

   • Measures the duration of events.

   • Conversions: 1 h = 60 min, 1 min = 60 s

   • Example: If a lesson starts at 9:15 am and ends at 10:40 am, what is the duration?
     10:40 − 9:15 = 1 h 25 min

5. Money

   • Ghanaian Cedis (GHS) used in transactions.

   • Example: If a student pays GHS 7.50 for a book and GHS 5.25 for a pen, total spent = 7.50 + 5.25 = GHS 12.75

Worked Examples (5)

1. Convert 5.8 km to metres.
   Solution: 1 km = 1000 m
   5.8 × 1000 = 5800 m

2. A water tank holds 7.2 L. Convert to mL.
   Solution: 7.2 × 1000 = 7200 mL

3. A sugar packet weighs 0.75 kg. Convert to grams.
   Solution: 0.75 × 1000 = 750 g

4. A football match lasted 1 hour 50 minutes. Convert to minutes.
   Solution: 1 h = 60 min → 60 + 50 = 110 min

5. A student has GHS 12.50 and spends GHS 4.25. How much is left?
   Solution: 12.50 − 4.25 = GHS 8.25

2. Angles, Shapes, Symmetry

Geometry is the study of shapes and angles.

A. Angles

• Definition: An angle is formed when two lines meet at a point (vertex).

• Units: Degrees (°)

• Types of Angles:

  • Acute: < 90°

  • Right: = 90°

  • Obtuse: > 90° but < 180°

  • Straight: = 180°

  • Reflex: > 180° but < 360°

Formulas

• Angles around a point = 360°

• Angles on a straight line = 180°

Examples

1. Angle A = 35°, find its complement.
   Solution: Complement = 90 − 35 = 55°

2. Angle B = 120°, find its supplement.
   Solution: Supplement = 180 − 120 = 60°

3. Two angles on a straight line are 70° and x. Find x.
   Solution: 70 + x = 180 → x = 110°

4. Angles in a triangle sum = 180°. Two angles 50° and 60°, find third.
   Solution: 180 − (50 + 60) = 70°

5. Angles around a point = 360°. Three angles 90°, 120°, x. Find x.
   Solution: 90 + 120 + x = 360 → x = 150°

B. Shapes

• 2D Shapes: Triangle, Quadrilateral, Circle, Rectangle, Square, Pentagon, Hexagon

• 3D Shapes: Cube, Cuboid, Cylinder, Sphere, Cone, Pyramid

Properties

• Triangle: 3 sides, sum of angles = 180°

• Quadrilateral: 4 sides, sum of angles = 360°

• Circle: 360° around center, πr² = area, 2πr = circumference

C. Symmetry

• Definition: Symmetry is when a shape can be divided into identical halves.

• Types:

  • Line (Reflection) Symmetry

  • Rotational Symmetry

Examples

1. Square: 4 lines of symmetry, rotational symmetry 90°

2. Rectangle: 2 lines of symmetry, rotational symmetry 180°

3. Equilateral Triangle: 3 lines of symmetry

4. Circle: infinite lines of symmetry

5. Letter A: 1 vertical line of symmetry

3. Triangles, Quadrilaterals, Circles

A. Triangles

• Types by sides: Equilateral, Isosceles, Scalene

• Types by angles: Acute, Right, Obtuse

• Properties

  • Sum of interior angles = 180°

  • Exterior angle = sum of opposite interior angles

Examples

1. Right triangle with angles 90°, 30°, find third angle.
   Solution: 180 − (90 + 30) = 60°

2. Scalene triangle with sides 5 cm, 7 cm, 8 cm. Perimeter?
   Solution: 5 + 7 + 8 = 20 cm

3. Equilateral triangle side 6 cm. Area?
   Solution: Area = (√3/4) × a² = (√3/4) × 36 ≈ 15.59 cm²

4. Right triangle sides 3 cm, 4 cm, find hypotenuse.
   Solution: c² = a² + b² → c² = 9 +16 = 25 → c = 5 cm

5. Exterior angle 110°, find opposite interior angles.
   Solution: 110 = sum of opposite angles

B. Quadrilaterals

• Types: Square, Rectangle, Parallelogram, Trapezium, Rhombus

• Properties:

  • Sum of interior angles = 360°

  • Opposite sides equal in parallelogram

  • Diagonals bisect each other in rhombus

Examples

1. Rectangle 8 m × 5 m, perimeter?
   Solution: 2(8 + 5) = 26 m

2. Rhombus diagonals 6 cm and 8 cm, area?
   Solution: Area = (d1 × d2)/2 = (6 × 8)/2 = 24 cm²

3. Trapezium bases 10 cm, 6 cm, height 4 cm, area?
   Solution: Area = ½(h)(b1+b2) = ½(4)(10+6)=32 cm²

4. Square side 7 cm, area?
   Solution: 7 × 7 = 49 cm²

5. Parallelogram base 12 cm, height 5 cm, area?
   Solution: 12 × 5 = 60 cm²

C. Circles

• Formulas:

  • Circumference: C = 2πr or πd

  • Area: A = πr²

Examples

1. Circle radius 7 cm, circumference?
   Solution: C = 2πr = 2 × 3.1416 × 7 ≈ 43.98 cm

2. Circle radius 5 cm, area?
   Solution: A = πr² = 3.1416 × 25 ≈ 78.54 cm²

3. Circle diameter 10 cm, circumference?
   Solution: C = πd = 3.1416 × 10 ≈ 31.416 cm

4. Circle radius 3 cm, find diameter.
   Solution: d = 2r = 6 cm

5. Circle area 50.24 cm², find radius.
   Solution: A = πr² → r² = 50.24 / 3.1416 ≈ 16 → r = 4 cm

4. Perimeter, Area, Volume

A. Perimeter

• Distance around a 2D shape

• Formula examples:

  • Rectangle: P = 2(l + w)

  • Square: P = 4 × side

  • Triangle: P = a + b + c

Examples

1. Square 6 cm → P = 4 × 6 = 24 cm

2. Rectangle 5 × 8 → P = 2(5+8) = 26 cm

3. Triangle 3,4,5 → P = 12 cm

4. Parallelogram base 7, side 4 → P = 2(7+4) = 22 cm

5. Trapezium sides 6,8,5,7 → P = 26 cm

B. Area

• Space enclosed in 2D shape

• Examples:

  • Square: A = s²

  • Rectangle: A = l × w

  • Triangle: A = ½(b × h)

  • Trapezium: A = ½(h)(b1 + b2)

  • Circle: A = πr²

Worked Examples (5)

1. Rectangle 6 × 9 → A = 54 cm²

2. Square 5 → 5² = 25 cm²

3. Triangle base 10, height 6 → ½(10×6)=30 cm²

4. Trapezium b1=8, b2=6, h=5 → ½(14×5)=35 cm²

5. Circle r=4 → 3.1416 × 16 ≈ 50.27 cm²

C. Volume

• Space occupied by 3D shape

• Formulas:

  • Cube: V = s³

  • Cuboid: V = l × w × h

  • Cylinder: V = πr²h

Worked Examples (5)

1. Cube side 3 cm → V = 3³ = 27 cm³

2. Cuboid 4 × 5 × 6 → V = 120 cm³

3. Cylinder r=3, h=7 → V = 3.1416 × 9 × 7 ≈ 197.92 cm³

4. Cube side 5 → V = 125 cm³

5. Cylinder r=2, h=10 → V = 3.1416 × 4 × 10 ≈ 125.66 cm³