8.3 Perimeter and Area 477 If we were to cut off the red portion of the parallelogram on the left, Fig. 8.29(a), and attach it to the right side of the figure, the resulting figure would be a rectangle, Fig. 8.29(b). Since the area of the rectangle is b h, × the area of the parallelogram is also b h. × Area of a Parallelogram A b h = × b b (a) (b) h h Figure 8.30 Consider the triangle with height, h, and base, b, shown in Fig. 8.30(a). Using this triangle and a second identical triangle, we can construct a parallelogram, Fig. 8.30(b). The area of the parallelogram is bh. The area of the triangle is one-half that of the parallelogram. Therefore, the area of the triangle is base height. 1 2 × × Area of a Triangle = A bh 1 2 Now consider the trapezoid shown in Fig. 8.31(a). We can partition the trapezoid into two triangles by drawing diagonal DB, as in Fig. 8.31(b). One triangle has base AB (called b2 ) with height DE, and the other triangle has base DC (called b1 ) with height FB. Note that the line used to measure the height of the triangle need not be inside the triangle. Because heights DE and FB are equal, both triangles have the same height, h. The area of triangles DCB and ADB are b h 1 2 1 and b h, 1 2 2 respectively. The area of the trapezoid is the sum of the areas of the triangles, + b h b h, 1 2 1 1 2 2 which can be written + h b b ( ). 1 2 1 2 b1 s2 s1 D A B C B C h (a) (b) b2 b1 A E E F D h h b2 Figure 8.31 Area of a Trapezoid = + A h b b ( ) 1 2 1 2 Did You Know? Area Versus Perimeter Italy New Mexico Which geographic region, Italy or New Mexico, do you think has the larger land area? As a matter of fact, the land areas of Italy and New Mexico are very similar: Italy’s area is 116,347 mi ,2 and New Mexico’s area is 121,697 mi .2 Italy’s perimeter, 5812 mi, is almost five times greater than New Mexico’s 1200 mi, however. Following is a summary of the perimeters and areas of selected figures. Perimeters and Areas Triangle Square Rectangle b (or s3) s1 h s2 s w l P s s s s b A bh ( ) 1 2 3 3 1 2 = + + = = P s A s 4 2 = = P l w A lw 2 2 = + = ( Continued )
RkJQdWJsaXNoZXIy NjM5ODQ=