Area of a Triangle

Figure 1-6.-Area of a rectangle. Figure 1-7.-Area of a triangle. Area of a Rectangle Figure 1-6 shows a rectangle measuring 10 ft by 8 ft, divided up into units of area measure, each consisting of 1 sq ft. If you were to count the units, one after the other, you would count a total of 80 units. However, you can see that there are 8 rows of 10 units, or 10 rows of 8 units. Therefore, the quickest way to count the units is simply to multiply 10 by 8, or 8 by 10. You could call the 8-ft dimension the width and the 10-ft dimension the length, in which case you would say that the formula for determining the area of a rectangle is the width times the length, or A = w1. Or, you could call the 10-ft dimension the base and the 8-ft dimension the altitude (meaning height), in which case your formula for area of a rectangle would be A = bh. Area of a Triangle Figure 1-7 shows a triangle consisting of one- half of the rectangle shown in figure 1-6. It is obvious that the area of this triangle must equal one-half of the area of the corresponding rectangle, and the fact that it does can be demonstrated by geometrical proof. Therefore, since the formula for the area of the rectangle is A = bh, it follows that the formula for the triangle is A = 1/2bh. The triangle shown in figure 1-7, because it is half of a corresponding rectangle, contains a right angle, and is therefore called a right triangle. In a right triangle the dimension h corresponds to the length of one of the sides. The triangle shown in figure 1-8, however, is a scalene triangle, so-called because no two sides are equal. Classification of triangles will be discussed later in this chapter. Now, a perpendicular CD drawn from the apex of the triangle (from angle C) divides the triangle into two right triangles, AADC and ABDC. The area of the whole triangle equals the sum of the areas of AADC and ABDC. The area of AADC equals 1/2 (AD)(DC), and the area of ABDC equals 1/2(DB)(DC). Therefore, the area of the whole triangle equals But since AD + DB = AB, it follows that the area of the whole triangle equals Figure 1-8.-Triangle. 1-10