Find the area from two sides and their included angle: S = (1/2)bc sin A
Example ①
In △ABC, find the area S when b = 6, c = 4, and A = 30°.
Solution: S = (1/2)×6×4×sin 30° = (1/2)×6×4×(1/2) = 6
In △ABC, find the area S when b = 6, c = 4, and A = 30°.
Solution: S = (1/2)×6×4×sin 30° = (1/2)×6×4×(1/2) = 6
Example ② (Heron's Formula)
Find the area of a triangle with sides a = 5, b = 6, and c = 7.
Solution: s = (5+6+7)/2 = 9. S = √(9×4×3×2) = √216 = 6√6
Find the area of a triangle with sides a = 5, b = 6, and c = 7.
Solution: s = (5+6+7)/2 = 9. S = √(9×4×3×2) = √216 = 6√6
Key Point
- Area: S = (1/2)bc sin A
- Heron's formula: S = √{s(s-a)(s-b)(s-c)} (s = (a+b+c)/2)
Practice Problems
- In △ABC, find the area S when b = 8, c = 5, and A = 60°.
Solutions and Explanations
- Solution: 10√3. S = (1/2)×8×5×(√3/2) = 10√3.