the (1/2)ab sin C area formula

The familiar area of a triangle, one-half base times height, needs the height — but in many real triangles the height is exactly the thing you have not measured. The (1/2)ab sin C formula sidesteps that: it computes the area directly from two sides and the angle wedged between them, with no perpendicular to drop.

If a triangle has two sides of lengths a and b meeting at an angle C between them, its area is Area = (1/2) x a x b x sin C. Here is why it is just the old formula in disguise: take side a as the base; the height of the triangle (the perpendicular distance from the opposite vertex down to that base) is b x sin C, because that vertex sits at the end of side b which is tilted up at angle C. Substituting into (1/2) x base x height gives (1/2) x a x (b x sin C), which is the formula. It needs the SAS data — two sides and their included angle — and nothing else.

This connects area to the law of sines and to Heron's formula as different lenses on the same triangle, and it is the everyday tool whenever you know two sides and the angle between them but not the height. The one thing to get right is that C must be the angle BETWEEN the two sides you are multiplying; using a side and a non-included angle gives the wrong answer.