Five key area expressions: (1) Delta = $\frac{1}{2}$baseheight — the basic formula. (2) Delta = $\frac{1}{2}$absinC — from two sides and included angle (most commonly used in trig). (3) Delta = sqrt(s(s-a)(s-b)(s-c)) — Heron's formula from three sides, where s = $\frac{a+b+c}{2}$. (4) Delta = rs — from inradius and semi-perimeter. (5) Delta = $\frac{abc}{4R}$ — from three sides and circumradius. Each formula is suited to different given data: use (2) for SAS, (3) for SSS, (4) when the inradius is known or sought, and (5) when the circumradius is involved. Additional formula: Delta = $\frac{1}{2}$|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| for coordinate geometry, connecting triangle properties to the coordinate plane.