PLANIMETRY. 9 
tude; it is, therefore, equal to a parallelogram whose base is equal to half 
the sum of the parallel sides, and whose altitude is equal to their perpendi- 
cular distance from each other (fig. 50). A rhombus, whose diagonals are 
perpendicular to each other, will be four times as large as a right angled 
triangle, which has for its two legs half the diagonals of the rhombus 
(fig. 51). The areas of two parallelograms as well as of two triangles of 
the same base, are to each other as their altitudes; of the same altitude, as 
their bases; and generally, parallelograms are to each other as the products 
of their bases by their altitudes. The areas of two squares are to each 
other as the squares of their sides. The areas of two similar triangles are 
to one another as the squares of their homologous or similarly situated sides 
(fig. 53); the same is true generally with regard to the areas of two similar 
figures. If on the three sides of a right angled triangle, three similar figures, 
triangles or any others, be constructed, the figure on the hypothenuse will 
be equivalent to the sum of those on the two legs (pl. 3, fig. 54). A par- 
ticular case of this proposition is known as the Pythagorean: the square 
described upon the hypothenuse is equivalent to the sum of the squares 
described on the other two sides. 
As the unit of measure for the determination of the superficial relations 
of figures, we use a square whose side is equal to the unit of length, which, 
therefore, according to the length of the side, is called a square foot, a square 
inch, &c. To ascertain how many times one square is contained in 
another, it is necessary to find out how many times the side of the one is 
contained in that of the other, and the number thus obtained multiplied 
by itself; hence a square foot contains not 10 or 12 square inches, but 100 
or 144, according to the number of inches, 10 or 12, into which the foot is 
divided, &c. The area of a square may thus be found, by measuring one 
of its sides and then multiplying the number expressing its length by 
itself. Hence we are accustomed to call the product of a number by itself, 
or the second power, its square. The area of a parallelogram is found by 
multiplying the base by the altitude (expressed in the same unit of measure) ; 
that of a triangle by multiplying the base by half the height. or the height 
by half the base ; that of a trapezoid by multiplying half the sum of the 
parallel sides by their perpendicular distance ; that of a regular polygon by 
multiplying its circumference or perimeter by half the perpendicular let fall 
from the centre on one of the sides ; that of an irregular polygon by divid- 
ing it by diagonals into triangles, whose areas must be separately ascer- 
tained and added together. 
By the assistance of the preceding propositions, many problems relative 
to the changing and dividing of figures may be solved. A few of these 
problems are the following :—1. To change a triangle into a parallelogram 
of equai area, or the contrary (fig. 55). 2. To change the triangle abc 
into another of equal area, and with a given side be (fig. 56). 3. To 
change a parallelogram into a rhombus of a given side cf (fig. 57), or of a 
given angle m (fig. 58). 4. To change a given triangle abc into an equi- 
lateral triangle (fig. 59). 5. To change a quadrilateral abcd into a trian- 
gle (fig. 60). 6. To change a given figure into another of a prescribed 
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