# JEE Main & Advanced Mathematics Trigonometric Equations Area of Triangle

## Area of Triangle

Category : JEE Main & Advanced

Let three angles of $\Delta ABC$ are denoted by $A,\,\,B,\,\,C$ and the sides opposite to these angles by letters $a,\,\,b,\,\,c$ respectively.

(1) When two sides and the included angle be given :

The area of triangle ABC is given by,

$\Delta =\frac{1}{2}bc\sin A=\frac{1}{2}ca\sin B=\frac{1}{2}ab\sin C$

i.e., $\Delta =\frac{1}{2}$ (Product of two sides) $\times$ sine of included angle

(2) When three sides are given :

Area of $\Delta ABC=\,\Delta =\sqrt{s(s-a)(s-b)(s-c)}$

where semiperimeter of triangle $s=\frac{a+b+c}{2}$

(3) When three sides and the circum-radius be given :

Area of triangle$\Delta =\frac{abc}{4R}$, where R be the circum-radius of the triangle.

(4) When two angles and included side be given :

$\Delta =\frac{1}{2}{{a}^{2}}\frac{\sin B\sin C}{\sin (B+C)}=\frac{1}{2}{{b}^{2}}\frac{\sin A\sin C}{\sin (A+C)}=\frac{1}{2}{{c}^{2}}\frac{\sin A\sin B}{\sin (A+B)}$

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