Integrated Math 3 / Unit 6: Trigonometric Identities and Laws
6.5 Area of a Triangle
Notes
- Using two sides and the included angle: $\displaystyle \text{Area} = \frac{1}{2}ab\sin C$.
- Example: $a = 10$, $b = 12$, $C = 30^\circ$ gives $\displaystyle \frac{1}{2}(10)(12)(0.5) = 30$.
- Heron's formula (three sides): $\displaystyle s = \frac{a + b + c}{2}$ and $\displaystyle \text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$.
- Example: sides 13, 14, 15 give $s = 21$ and area $\displaystyle \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84$.
Practice
Two sides of a triangle are 10 and 12 with an included angle of $30^\circ$. What is its area?
$\displaystyle \frac{1}{2}(10)(12)\sin 30^\circ = 60(0.5) = 30$.
A triangle has sides 13, 14 and 15. What is its area?
$s = 21$, so the area is $\displaystyle \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84$.
A triangle has sides 9 and 14 with an included angle of $150^\circ$. What is its area?
$\displaystyle \frac{1}{2}(9)(14)\sin 150^\circ = 63(0.5) = 31.5$.
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