Proof of the area formula

The trigonometric formula for the area is derived from the familiar “base times height divided by two”, expressing the height with the sine.

Proof

The area of a triangle is $S = \tfrac12\,\text{base}\cdot\text{height}$. Take $c = AB$ as the base and the perpendicular from $C$ to $AB$ as the height: this perpendicular has length $b\sin\widehat{A}$ (it is the side opposite $\widehat{A}$ in the right-angled triangle formed by the perpendicular and the side $AC$). Hence $S = \tfrac12\,c\cdot b\sin\widehat{A}. \qquad \blacksquare$

Links

Topics: Triangle trigonometry
Concepts: Triangle area
Functions: Sine
Skills: Proving