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PART II.

OF SUPERFICIAL MENSURATION, OR THE MENSURATION OF PLANE FIGURES.

SECTION I.

OF THE AREAS OF RIGHT-LINED AND CIRCULAR

THE

FIGURES.

HE measure of a plane figure is called its area. By the menfuration of plane figures is determined the extenfion of bodies as to length and breadth; fuch as the quantities of lands, and the works of many artificers.

Plane figures, and the furfaces of bodies, are meafured by fquares; as fquare inches, or fquare feet, or fquare yards, &c; that is, fquares whofe fides are inches, or feet, or yards, &c. Our leaft fuperficial meafure is the fquare inch, other fquares being taken from it according to the proportion in the following table.

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PROBLEM I.

To find the Area of a Parallelogram, whether it le a Square, a Rectangle, a Rhombus, or a Rhomboid.

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* Multiply the length by the height or perpendicular breadth, and the product will be the area. That is, AB X AC the area.

Note. Because the length of a fquare is equal to its height, its area will be found by multiplying the fide by itself.-That is AB X AB or AB is the area of the fquare.

*

DEMONSTRATION.

EX

For, let ABCD be a rectangle; and let A its length AB and CD, and its breadth AD and BC, be each divided into as many equal parts, as is expreffed by the number of times they contain the lineal measuring unit; and let all the oppofite points of divifion be connected by right D lines. Then, it is evident that, these

B

lines divide the rectangle into a number of fquares, each equal to the fuperficial meafuring unit; and that the number of thefe fquares is equal to the number of lineal meafuring units in the ' length, as often repeated as there are lineal measuring units in the breadth, or height; that is, equal to the length drawn into the breadth. But the area is equal to the number of fquares or fuperficial measuring units; and therefore the area of a rectangle is equal to the product of its length and breadth.

Again, a rectangle is equal to an oblique parallelogram of an equal length and perpendicular height, by Euclid I. 36. Therefore the area of every parallelogram is equal to the product of its length and height. 2. E.D.

EXAMPLES.

1. What is the area of a parallelogram whofe length is 12.25 chains, and its height 8.5 chains?

12.25 length
8.5 breadth

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Note. Four roods are equal to an acre, and therefore 40 perches, or fquare poles, make a rood.

Ex. 2. What is the area of a fquare whofe fide is 35.25 chains? Anf. 124ac. ir. ip. Ex. 3. What is the area of a rectangular board, whofe length is 12.5 feet, and breadth 9 inches? Anf. 9.375 fq. feet. Ex. 4. How many fquare yards of painting are in a rhombus, or a rhomboid, whofe length is 37 feet, and perpendicular breadth 5.25 feet?

RULE II.

Anf. 21.58.

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As radius (viz. fine of 90° or tang. of 45°)
Is to the fine of any angle of a parallelogram
So is the product of the fides including the angle:
To the area of the parallelogram.

That

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* That is, AB X BC X nat. fine of the angle в = the area.

Note. Because the angles of a fquare and rectangle are each 90°, whofe fine is 1, this rule, for them, is the fame as the former.

EXAMPLES.

1. What is the area of a rhomboides whofe length is 36 feet, flope height 25.5 feet, and one of the lefs angles 58°?

(Rad.) 1: 8480481 (nat. fine of 58°) :: 918 (= 25.5 × 36): 778-5081558 the area.

Or, to use the Logarithms.

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Ex. 2. What is the area of a parallelogram whose angle is 90°, and the including fides 20 and 12.25 chains? Anf. 245 acres. Ex. 3. What is the area of a rhombus, each of whose fides is 21 feet 3 inches, and each of the less angles 53° 20' ? Anf. 362.208757 feet. Ex. 4. How many acres are in a rhomboides whose less angle is 30°, and the including fides 25.35 and 104 chains? Anf. 13 ac. 29.12 per.

RULE

* DEMONSTRATION.

For, having drawn the perpendicular AP, the area, by the first rule, is AP X BC; but as rad. 1 (s. ≤P) : S. 4B ;; AB : AP =

S. LB X AB; therefore AP X BC area; or I S. LB :: AB X BC: S. of the parallelogram. 2. E.D.

BC X S. 4B X AB, is the
LE X AB X BC= the area

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To the fine of the angle which the diagonals of a

parallelogram make with each other

So is the product of the diagonals

To double the area.

That is,

AB X CD X nat. s. P

2

the area.

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Note. Because the diagonals of a fquare and rhombus interfect at a right angle, whofe fine is 1, therefore half the product of their diagonals is the area.

That is, AB in the fquare, and AB XCD in the rhombus, is the area.

EXAMPLES.

1. How many fquare yards of pavement are in a fquare whofe diagonal is 27 feet 6 inches?

27.5

27°5

1375

1925

550

2)756.25

9)378.125 feet

42.013 yards

Ex.

*This rule is common to all quadrilaterals, and is proved at

cafe 2 of prob. 3.

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