Key Learning Objectives
- Calculate triangle areas using Area = 1/2 × base × perpendicular height.
- Understand how dropping a perpendicular height creates right-angled triangles.
- Follow the trigonometric proof connecting height h = b × sin(C) to Area = 1/2 a b sin(C).
Topic Summary & Core Notes
Master triangle area calculations from fundamental KS3 base × height ÷ 2 to the algebraic trigonometric proof for non-right angled triangles: Area = 1/2 a b sin(C).
Key Rules, Formulas & Definitions
Standard Triangle Area
Area = ½ × base × perpendicular height [Area = ½ b h]
Trigonometric Triangle Area
Area = ½ × a × b × sin(C) (where C is the included angle between sides a and b)
Perpendicular Height Proof
sin(C) = Opposite ÷ Hypotenuse = h ÷ b → h = b × sin(C)
2 Step-by-Step Worked Examples
Worked Example 1
Worked Example 1: Standard Area Calculation
Question: A triangle has a base of 14 cm and a perpendicular height of 9 cm. Calculate its area.
Step-by-Step Solution:
- Step 1: Write formula: Area = ½ × base × height.
- Step 2: Substitute values: Area = 0.5 × 14 cm × 9 cm.
- Step 3: Calculate: Area = 7 × 9 = 63 cm².
Final Answer: 63 cm²
Senior Examiner Insight: Always check units are cm² or m² for area calculations.
Worked Example 2
Worked Example 2: Applying the 1/2 ab sin(C) Formula
Question: A non-right triangle has sides a = 8 cm, b = 10 cm, and an included angle C = 30°. Calculate its area (given sin 30° = 0.5).
Step-by-Step Solution:
- Step 1: Write formula: Area = ½ × a × b × sin(C).
- Step 2: Substitute values: Area = 0.5 × 8 × 10 × sin(30°).
- Step 3: Calculate: Area = 0.5 × 8 × 10 × 0.5 = 20 cm².
Final Answer: 20 cm²
Senior Examiner Insight: The angle C MUST be the 'included angle' directly between sides a and b.
Common Pitfalls & Examiner Warnings
Common Mistake: Using slanted side length instead of perpendicular height
Standard formula Area = 1/2 b h requires perpendicular (90°) vertical height.
Common Mistake: Using a non-included angle in 1/2 ab sin(C)
The angle must be enclosed directly between the two given side lengths.
Common Mistake: Forgetting to divide by 2
Failing to multiply by 0.5 gives the area of a parallelogram instead of a triangle.
Interactive Self-Check Quiz (3 Questions)
Question 1 of 3
What is the area of a right-angled triangle with base 6 cm and height 8 cm?
Click to Reveal Answer & Mark Scheme
Correct Answer: B) 24 cm²
Detailed Explanation: Area = 0.5 × 6 × 8 = 24 cm².
Mark Scheme & Scoring Points: 1 mark for 24 cm².
Question 2 of 3
In the formula Area = 1/2 a b sin(C), what is the relationship between angle C and sides a and b?
Click to Reveal Answer & Mark Scheme
Correct Answer: A) C must be the included angle located between sides a and b.
Detailed Explanation: The formula requires the angle sandwiched between the two known side lengths.
Mark Scheme & Scoring Points: 1 mark for included angle between sides a and b.
Question 3 of 3
If sin(30°) = 0.5, what is the area of a triangle with sides 12 cm and 10 cm enclosing a 30° angle?
Click to Reveal Answer & Mark Scheme
Correct Answer: B) 30 cm²
Detailed Explanation: Area = 0.5 × 12 × 10 × 0.5 = 30 cm².
Mark Scheme & Scoring Points: 1 mark for 30 cm².
Created by Fiaraz Iqbal
Former Headteacher & Senior Science / Maths Examiner
Fiaraz produces structured video walkthroughs, Tier 3 literacy packs, and exam mark scheme breakdowns to help secondary students achieve top grades.
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