Sa consideram triunghiul ABC, cu notatiile uzuale: AB=c, BC=a, AC=b si [tex] h_{A}, h_{B} ~si~ h_{C} [/tex] lungimile inaltimilor din A, B, respectiv C.
[tex] \frac{a*b*sinC}{2}= \frac{a*b* \frac{ h_{A} }{b} }{2}= \frac{a*h_{a} }{2}= A_{ABC} \\ \frac{b*c*sinA}{2}= \frac{b*c* \frac{ h_{B} }{c} }{2}= \frac{b* h_{B} }{2}= A_{ABC} \\ \frac{a*c*sinB}{2}= \frac{a*c* \frac{ h_{C} }{a} }{2}= \frac{c* h_{C} }{2}= A_{ABC} [/tex]
Pentru a calcula aria triunghiului echilateral, folosim ceea ce am calculat anterior.
Notez cu l latura.
[tex]A= \frac{l*l*sin60}{2}= \frac{ l^{2}* \frac{ \sqrt{3} }{2} }{2}= \frac{ l^{2} \sqrt{3} }{4} .[/tex]
