Area from two sides and the included angle

In \(\triangle ABC\), side \(a\) is opposite angle \(A\), side \(b\) opposite \(B\), and side \(c\) opposite \(C\). Two sides and their included angle determine area: \(\text{Area}=\dfrac12 ab\sin C\). Dropping a perpendicular gives height \(b\sin C\) to base \(a\).

The same formula works for an obtuse included angle, because the external height uses its acute supplement and the two sines are equal.

Example

Two sides are 7 cm and 10 cm with included angle \(30^\circ\). Area \(=\dfrac12(7)(10)\sin30^\circ=17.5\text{ cm}^2\). With included angle \(150^\circ\), the area is also 17.5 cm², although the triangles are not congruent.

The angle must lie between the two lengths you substitute. Using an angle opposite one of those sides in this formula gives the wrong area.