$A = \tfrac{1}{2}bh$ is honest and useless in the field, because nobody surveys a perpendicular height — you cannot drop a plumb line from a fence corner to a boundary that is not there. What you CAN measure is two sides and the angle between them.

So build the height instead. With sides $a$ and $b$ meeting at $C$, the height onto $b$ is $a\sin C$ — straight out of SOH. Substitute: $A = \tfrac{1}{2}\,b\,(a\sin C) = \tfrac{1}{2}ab\sin C$. Read it aloud: area equals one half a b sine C. Nothing new was invented; the sine simply does the job the missing height would have done.

$C$ must be the INCLUDED angle, the one both sides touch — feed it an angle from somewhere else in the triangle and the formula answers a question about a different triangle entirely. Two sanity rails: at $C = 90^\circ$, $\sin C = 1$ and the formula collapses to $\tfrac{1}{2}ab$, the plain right-triangle area; and the answer wears SQUARE metres, because two lengths multiplied always do.