Triangle Area Trigonometry

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📐 Part 1: Support Level - Guided Practice

WALT: Apply trigonometric relationships to calculate the area of triangles using ½ ab sin C

Formula: Area = ½ × a × b × sin C (where a and b are two sides, and C is the included angle)

1. Complete this worked example:

Triangle with sides a = 8 cm, b = 6 cm, and included angle C = 30°

Step 1: Write the formula: Area = ½ × a × b × sin C

Step 2: Substitute values: Area = ½ × ____ × ____ × sin(____°)

Step 3: Calculate sin(30°) = ________

Step 4: Complete calculation: Area = ½ × 8 × 6 × 0.5 = ________ cm²

2. Now try this one with hints:

Triangle with sides a = 10 m, b = 12 m, and included angle C = 60°

Hint: sin(60°) = 0.866

3. Which angle should you use in the formula ½ ab sin C?

Any angle in the triangle

The largest angle

The angle between the two given sides

The smallest angle

📊 Part 2: Mainstream Level - Standard Problems

4. Calculate the area of a triangle with sides 15 cm and 20 cm, with an included angle of 45°.

5. Find the area of a triangle where two sides are 9 m and 14 m, and the angle between them is 75°.

6. A triangle has sides of 18 cm and 25 cm with an included angle of 120°. Calculate its area.

7. Which of these scenarios would require the sine area formula? (Tick all that apply)

A right-angled triangle with base and height given

Any triangle with two sides and the included angle known

An obtuse triangle with two sides and included angle given

A triangle with all three sides known but no angles

🚀 Part 3: Extension Level - Multi-Step Problems

8. A surveyor needs to find the area of a triangular block of land. Two boundary walls meet at 70°, with lengths of 45 m and 60 m. Calculate the area in square metres, then convert to hectares. (1 hectare = 10,000 m²)

9. Challenge Problem: In triangle ABC, side AB = 16 cm, angle A = 35°, and angle B = 80°. Find the area of the triangle.

Hint: You'll need to find angle C first, then use the sine rule to find another side before calculating the area.

10. Real-World Application: An architect is designing a triangular roof section. Two roof beams of 8 m and 12 m meet at an angle of 110°. If roofing material costs $85 per square metre, what will the material cost for this section?

11. Explain in your own words when you would use the formula ½ ab sin C instead of other area formulas: