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