GCSE Maths - Essential Formulas for Exam Success

Introduction

• This video covers essential crossover formulas needed for both foundation and higher papers in GCSE Maths.
• Separate video available for higher paper-specific formulas.
• Surface-level examples provided for each formula.

Areas

Area of a Rectangle

• Formula: Area = Length × Width
• Example: For a rectangle with length 3 cm and width 8 cm, Area = 3 × 8 = 24 cm²

Area of a Triangle

• Formula: Area = ½ × Base × Height
• Example: For a triangle with base 10 cm and perpendicular height 7 cm, Area = ½ × 10 × 7 = 35 cm²

Area of a Trapezium

• Formula: Area = ½ × (a + b) × Height
• Example: For a trapezium with a = 8 cm, b = 10 cm, and height = 5 cm, Area = ½ × (8 + 10) × 5 = 45 cm²

Area of a Parallelogram

• Formula: Area = Base × Height
• Example: For a parallelogram with base = 8 cm and height = 5 cm, Area = 8 × 5 = 40 cm²

Area of a Circle

• Formula: Area = πr²
• Example: For a circle with radius 4 cm, Area = π × 4² ≈ 50.3 cm²

Circles

Circumference of a Circle

• Formula: Circumference = π × Diameter or π × 2r
• Example: For a circle with diameter 6 cm, Circumference = π × 6 ≈ 18.8 cm

Volumes

Volume of a Prism

• Formula: Volume = Area of Cross-Section × Length
• Example: For a triangular prism with base 6 cm, height 4 cm, and length 10 cm:
  • Area of Cross-Section = ½ × 6 × 4 = 12 cm²
  • Volume = 12 × 10 = 120 cm³

Volume of a Cylinder

• Formula: Volume = πr² × Height
• Example: For a cylinder with radius 3 cm and height 8 cm:
  • Volume = π × 3² × 8 ≈ 226.2 cm³

Pythagoras' Theorem

• Formula: a² + b² = c²
• Example: For a right-angled triangle with sides 3 and 4 cm:
  • 3² + 4² = c² → c² = 25 → c = 5 cm
• Note: Can also find shorter sides using subtraction.

Trigonometric Ratios

• Sine: sin θ = Opposite/Hypotenuse
• Cosine: cos θ = Adjacent/Hypotenuse
• Tangent: tan θ = Opposite/Adjacent

Example Using Sine

• Given: Opposite = 5 cm, Angle = 30°
• Formula: Hypotenuse = Opposite / sin θ → Hypotenuse = 5 / sin 30 ≈ 10 cm

Example Using Cosine

• Given: Hypotenuse = 8 cm, Angle = 30°
• Formula: Adjacent = cos θ × Hypotenuse → Adjacent = cos 30 × 8 ≈ 6.9 cm

Other Formulas

Density

• Formula: Density = Mass / Volume
• Example: For mass = 200 g and volume = 40 cm³:
  • Density = 200 / 40 = 5 g/cm³

Speed

• Formula: Speed = Distance / Time
• Example: For a car traveling 448 miles in 8 hours:
  • Speed = 448 / 8 = 56 mph

Pressure

• Formula: Pressure = Force / Area
• Example: For a force = 3500 N and area = 5 m²:
  • Pressure = 3500 / 5 = 700 N/m²

Conclusion

• These formulas are essential for both higher and foundation level GCSE Maths.
• Check out the next video for higher-tier only formulas.
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