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Area of Oblique Triangles

given 3 sides:

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$S=\frac{1}{2}(a+b+c)$

given two sides and the enclosed angle:

$A=\frac{1}{2}b\cdot c\cdot sin(A)$

$A=\frac{1}{2}a\cdot c\cdot sin(B)$

$A=\frac{1}{2}a\cdot b\cdot sin(C)$

Examples: Find the area of the triangle.

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$s=\frac{1}{2}(a+b+c)=\frac{1}{2}(45+51+38)=67$

$A=\sqrt{67(67-45)(67-51)(67-38)}$

$A=\sqrt{67(22)(16)(29)}=\sqrt{683936}=827\ in^{2}$

$A=\frac{1}{2}a\cdot b\cdot sin(c)$

$A=\frac{1}{2}(10)(7)\cdot sin(108\degree)$

$A=\frac{1}{2}(70)(0.951)$

$A=33.3\ cm^{2}$