14-4 Area and the Law of Sines

Objectives

To find the area of any triangle

To use the Law of Sines

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Recall from geometry that if you know three parts of a triangle then you can sometimes solve the triangle; that is, you can determine its complete shape. This is what the congruence statements SAS, ASA, AAS, SSS were all about.

Essential Understanding If you know two angles and a side of a triangle, you can use trigonometry to solve the triangle. If you know two sides and the included angle, you can find the area of the triangle.

The area of a triangle with base b and height h is 1 2 1 half bh. When you don't know h but you do know an angle measure, there may be another way to find the area.

Here's Why It Works The area of the triangle above is 1 2 1 half bh. The altitude h to side AC completes a right triangle. Thus, sin A = h c , sine eh equals , h over c . comma so h = h equals c sin A. Substituting, 1 2 1 half bh = 1 2 equals , 1 half bc sin A. You can derive 1 2 1 half ac sin B and 1 2 1 half ab sin C in a similar way.

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