Trigonometry / Unit 8: Oblique Triangles and Vectors
8.4 Area of a Triangle
Notes
There are two main area formulas for oblique triangles.
- SAS area: $\displaystyle \text{Area} = \frac{1}{2}ab\sin C$, where $C$ is the angle between sides $a$ and $b$.
- Heron's formula: with semiperimeter $\displaystyle s = \frac{a + b + c}{2}$, $\displaystyle \text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$.
- Example: sides 6 and 8 with a $30^\circ$ angle between them have area $\displaystyle \frac{1}{2}(6)(8)\sin 30^\circ = 12$.
Practice
A triangle has two sides of length 10 and 12 with a $30^\circ$ angle between them. What is its area?
$\displaystyle \frac{1}{2}(10)(12)\sin 30^\circ = 60 \cdot \frac{1}{2} = 30$.
Use Heron's formula to find the area of a triangle with sides 13, 14 and 15.
$s = 21$, so Area $\displaystyle = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84$.
What is the exact area of an equilateral triangle with side length 6?
$\displaystyle \frac{1}{2}(6)(6)\sin 60^\circ = 18 \cdot \frac{\sqrt{3}}{2} = 9\sqrt{3}$.
Ready for more? Take the Unit 8 test.