D

B

C

8. The area of a figure is the superficial contents contained within the line or lines, by which the figure is bounded. 9. A rectangle is said to be contained by any two of the straight lines which are adjacent to one of its angles. Thus the rectangle ABCD, is contained by the lines AB and BC, or by BC and CD, &c. For the sake of brevity, rectangles and parallelograms are often designated by two letters at the opposite corners as AC, or BD.

A

B

Note. The product of two lines AB, BC, is often called their rectangle. When it is said that two lines are multiplied together, it must always be understood that the number of linear units in one line, is multiplied by the numbers of linear units in the other. And when it is said that a surface is multiplied by a line, it is implied that the number of measuring units in the surface, is multiplied by the number of linear units in the line. (See Alg. Arts. 383—390.)

Parallelograms which have equal bases and equal altitudes are equivalent.

ses DC, FE will be situated both in one straight line parallel to AB.

Now, from the nature of parallelograms, (30. 1,) we have AD=BC, and AF BE; hence the triangles ADF, BCE, have the two sides AD, AF equal to BC, BE, each to each; and since DC=AB, (30. 1,) and FE=AB, hence DC=FE; (Ax. 1;) and if DC and EF be taken from DE, the remainders CE and FD will be equal. Therefore the triangles ADF, BCE are mutually equilateral and consequently equal. (10.1.)

But if from the quadrilateral ABED, we take away the triangle ADF, there will remain the parallelogram ABEF: and if from the same quadrilateral ABED, we take away the triangle CBE, there will remain the parallelogram ABCD. Hence these two parallelograms ABCD, ABEF, which have the same base and altitude, are equivalent. (Ax. 3.) Therefore, Parallelograms, &c.

Cor. Every parallelogram is equivalent to the rectangle which has the same base and the same altitude.

Every triangle ABC is half of the parallelogram ABCD, which has the same base and the same altitude.

For, since by hypothesis, ABCD is a parallelogram of which AC is a diagonal, therefore, (30. 1. Cor. 1,) the triangle ABC=ACD: hence the triangle ABC is equal to half the parallelogram ABCD. Therefore,

F

B

E

D

Every triangle is half of the parallelogram which has the same base and the same altitude.

Cor. 1. Hence a triangle ABC is half of the rectangle BCEF, which has the same base CB, and the same altitude

AO: for the rectangle BCEF is equivalent to the parallelogram ABCD. (1. 4. Cor.)

Cor. 2. All triangles, which have equal bases and altitudes, are equivalent.

Two rectangles ABCD, AEFD having the same altitude AD, are to each other as their bases, AB, AE.

vision erect a perpendicular to the base; seven partial rectangles will thus be formed, all equal to each other, because all have the same base and altitude. The rectangle ABCD will contain seven partial rectangles, while AEFD will contain four hence the rectangle ABCD : AEFD : : 7:4: but by hypothesis, 7: 4 :: AB: AE; therefore, (11. 3,) ABCD; AEFD: AB: AE.

The same reasoning may be applied to any other ratio as well as to that of 7 to 4. Hence,

Any two rectangles having the same altitude, are to each other as their bases.

Two rectangles ABCD, AEGF, are to each other as the products of the bases multiplied by the altitudes; that is, ABCD: AEGF:: AB×AD: AEXAF.

Having placed the two rectangles, so that the angles at A are vertical, produce the sides GE, CD till they meet in

tude AE, are to each other as their bases AD, AF: thus we have the two proportions,

ABCD AEHD::AB: AE,

AEHD AEGF :: AD: AF.

Multiplying the corresponding terms of these proportions together, omitting the mean term AEHD, (18. 3. Cor.,) we shall have the proportion,

ABCD AEGF:: AB× AD: AEX AF.

Hence,

Any two rectangles are to each other as the product of their bases multiplied by their altitudes.

Scholium. The product of the base by the altitude, may therefore be assumed as the measure of a rectangle.

But the more common and simple method is to assume the square as the measuring unit of surfaces; and that square has been adopted, whose side is the unit of length, or a linear unit. For example, if the length of the rectangle A is six units and its altitude three, then 6×3, or 18, repre

sents the number of superficial units or squares, contained in the rectangle,

A

and each side of these squares is equal to a linear unit of the same name as the square. If the side of the square is a linear inch, the rectangle contains 18 square inches; if the side of the square is a foot, the rectangle contains 18 square feet.

The area of a parallelogram ABCD, is equal to the product of its base AB by its altitude BE.

F D

E C

For the parallelogram ABCD is equivalent to the rectangle ABEF, which has the same base AB, and the same altitude BE; (1. 4. Cor. ;) but this rectangle is measured by ABX BE; (4.4. Sch. ;) therefore AB BE is equal to the area of the parallelogram ABCD. Hence,

A

B

The area of every parallelogram is equal to the product of its base by its altitude.

Cor. 1. The area of a square is equal to the square of one of its sides. For, if AB=BE, then AB×BE=AB2, BE2.

or

Cor. 2. Parallelograms of the same base are to each other as their altitudes; and parallelograms of the same altitude are to each other as their bases: for if A and B are the altitudes of two parallelograms, and C their base; then A×C is equal to the area of one, and BXC the other.

But

The area of a triangle is equal to the product of its base by half its altitude.

For, the triangle ABC is half of the parallelogram ABCE, which has the same base BC, and the same altitude AD: (2. 4:) but the surface of the parallelogram is equal to BC × AD;

E