While there is no simple formula to determine area for most quadrilaterals, and most polygons for that matter, we saw last section that for the special quadrilaterals parallelograms and trapezoids there are specific formulas for determining area. The area of a triangle, however, does. This is why it is so important that any polygon can be divided into a number of triangles. The area of a polygon is equal to the sum of the areas of all of the triangles within it.
The area of a triangle can be calculated in three ways. The most common expression for the area of a triangle is one-half the product of the base and the height (1/2AH). The height is formally called the altitude, and is equal to the length of the line segment with one endpoint at a vertex and the other endpoint on the line that contains the side opposite the vertex. Like all altitudes, this segment must be perpendicular to the line containing the side. The side opposite a given vertex is called the base of a triangle. Here are some triangles pictured with their altitudes.
Another way to calculate the area of a triangle is called Heron's Formula, named after the mathematician who first proved the formula worked. It is useful only if you know the lengths of the sides of a triangle. The formula makes use of the term semiperimeter. The semiperemeter of a triangle is equal to half the sum of the lengths of the sides. Heron's Formula states that the area of a triangle is equal to the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter of the triangle, and a, b, and c are the lengths of the three sides. The proof of Heron's Formula is rather complex, and won't be discussed here, but his formula works like a charm, especially if all that is known about a triangle is the lengths of its sides.
The third and final way to calculate the area of a triangle has to do with the angles as well as the side lengths of the triangle. Any triangle has three sides and three angles. These are known as the six parts of a triangle. Let the lengths of the sides of the triangle equal a, b, and c. If the vertices opposite each length are angles of measure A, B, and C, respectively, then the triangle would look like this: The area of such a triangle is equal to one-half the product of the length of two of the sides and the sine of their included angle. So in the triangle above, Area=1/2(ab sin(C)), or Area=1/2(bc sin(A)), or Area=1/2(ac sin(B)). The sine of an angle is a trigonometric property that you can learn more about in the Trigonometry SparkNote.
With these formulas, it is possible to calculate the area of a triangle any time you have any of the following information: 1) the length of the base and its altitude; 2) the lengths of all three sides; or 3) the length of two sides and the measure of their included angle. With these tools, it is possible to calculate the area of triangles, and, as we shall see, by summing the areas of triangles within a polygon, it is possible to calculate the area of any polygon.