Area of an Oblique Triangle
You don't need a height to find the area. If you know two sides and the included angle (SAS), the sine area formula gives the answer directly.
Introduction
The classic formula needs a height. But when you know SAS, sine gives you the height: . This leads to the elegant SAS area formula.
Past Knowledge
Triangle area = ½bh, SAS identification (7.1), sine function.
Today's Goal
Calculate the area of any triangle using two sides and the included angle.
Future Success
Heron's Formula (7.7) finds area when you only know the three sides — no angle needed.
Key Concepts
The SAS Area Formula
Any pair of sides with the included angle works:
Why Sine?
In a triangle with sides and included angle , the height from the vertex to side is . So .
Worked Examples
Direct SAS
Given: .
Answer: Area square units
Obtuse Included Angle
Given: .
Note: , so the formula works perfectly for obtuse angles too.
Answer: Area square units
AAS → Area
Given: . Find the area.
Step 1: . Find via Law of Sines:
Step 2: Now use SAS area with sides and included :
Answer: Area square units
Common Pitfalls
Using a Non-Included Angle
The angle must be between the two sides. Using an angle opposite one of the sides gives a wrong area. If you don't have the included angle, find it first.
Real-Life Applications
Real Estate — Lot Sizing
Irregularly shaped lots are common. A surveyor can measure two boundary lengths and the angle where they meet, then use the SAS area formula to calculate the exact lot area — no need to measure an altitude.
Practice Quiz
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