Introduction
Learn two powerful methods to find the area of any triangle: the sine formula for SAS, and Heron's formula for SSS.
Past Knowledge
You know the Law of Sines and Law of Cosines to solve triangles.
Today's Goal
We're applying the sine area formula and Heron's formula.
Future Success
Surveyors calculate land areas from measurements without needing perpendicular heights.
Area Formulas
Sine Area Formula (SAS)
Use when you know two sides and the included angle.
Heron's Formula (SSS)
Where (semi-perimeter)
Worked Examples
Example 1: Sine Area Formula
Find the area given , , .
Answer: square units
Example 2: Heron's Formula
Find the area given , , .
Step 1:
Step 2:
Answer: square units
Example 3: Real-World Problem (Advanced)
A triangular plot of land has sides 120m, 150m, and 200m. Find its area in hectares.
Step 1:
Step 2: m²
Step 3: Convert: hectares
Answer: ≈ 0.895 hectares
Common Pitfalls
Wrong angle in sine formula
The angle C must be the included angle between sides a and b.
Forgetting semi-perimeter
Heron's formula uses (half the perimeter), not the full perimeter.
Real-World Application
Land Surveying & Real Estate
Surveyors use these formulas to calculate the area of irregularly shaped plots of land. By measuring the sides, they can determine acreage for property valuations, tax assessments, and development planning.
Practice Quiz
Practice Quiz
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