Finding the Area of Any Triangle

From base and height to sine

You already know that triangle area can be found with $\frac{1}{2}\times \text{base}\times \text{height}$ . In this sketch, $C Q = a$ is the base and $C P = b$ .

         P
        /|
       / |
    b /  | h
     /   |
    /    |
   C-----D------Q
   <------ a ------>

Angle $C$ is between sides $a$ and $b$ . Dropping a perpendicular from $P$ to the base at $D$ gives the height $P D = h$ . In right triangle $C P D$ , $sin⁡C=PDCP=hb\sin C = \frac{PD}{CP} = \frac{h}{b}$ , so $b\sin C$ . Substituting that into the usual area formula gives

Finding the Area of Any Triangle Lesson

Finding the Area of Any Triangle Lesson