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Article 74. To measure a Sector.

Definition. A Sector is a Part of a Circle, contained between an Arch-Line and two Radii or Semidiameters of the Circle. Rule. Find the Length of Half the Arch, by Art. 72; then multiply this by the Radius or Semidiameter, and the Produc is the Area.

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72.333 Length of the Arch ABC, by Art. 72.

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Ex. 2. In the Sector ABCD, greater than a Semicircle, given the Radius AE or ED 56, the Chord BD of Half the Arch ABD 102, and the Chord BC of Half the Arch B C D 60, to find the Area of the Sector?

60 B C.

2

120

*102 fubtract.

3) 18

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120 add.

630

7056 Area of the Sector,

126 Length of the Arch BCD, by Art. 72.

Article 75. To find the Area of a Segment of a Circle. Definition. A Segment of a Circle is any Part of a Circle cut off by a right Line drawn crofs the Circle, which does not pass through the Center; for, if a right Line paffes through the Center, it is a Diameter, and divides the Circle into two equal Parts called Semicircles; but a Segment is always either greater or less than a Semicircle.

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Rule. By Art. 72, find the Length of the Arch-Line A BC, and, by Art. 73, the Diameter BF; then multiply Half the Chord of the Arch ABC by Half the Diameter, and the Product

F

is the Area of the Sector ABCE: Then find the Area of the Triangle A EC, whofe Bafe A C is 86, and perpendicular Height DE 17, found by fubtracting the verfed Sine BD from Half the Diameter; and the Area of the Triangle A E C, being fubtracted from the Area of the Sector ABCE, leaves the Area of the Segment A B C.

T 2

52 BC.

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Firft find the Area of the Sector ABCDE, by Art. 72, at the fecond Example; then find the Area of the Triangle .AED, by Art. 56; and, adding the Area of the Triangle to the Area of the Sector, that Sum is the Area of the Segment.

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The Arch-Line ABC being but Half the Arch of the Sector, multiply it by 40, the Radius, and the Product 3613.32 is the Area of the Sector; to which adding 714, the Area of the Triangle, their Sum 4327.32 is the Area of the Segment.

Article 76. To find the Area of an Ellipfis.

Definition. An Ellipfis, commonly called an Oval, is a Curve which returns into itself like a Circle, but has two different Diameters, one longer than the other.

Rule. Multiply the two Diameters of the Ellipfis together; then multiplying that Product by .78539, or .7854, this last Product is the Area of the Ellipfis.

In the Ellipfis ABCD, the tranfverfe or longeft Diameter BD is 44, and the conjugate or fhorteft Diameter AC 36, to find the Area?

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1244.0736 Area of the Ellipfis.

Article 77. To meafure a Parabola.

D

Definition. A Parabola is a Figure made by cutting a Cone [See the Definition of a Cone, Art. 81.] by a Plane parallel to one of the Sides: But the Figure will give the Learner a better Idea than any Definition.

Rule. Multiply the Bafe by the Height, and that Product by 2; then divide this laft Product by 3, and the Quotient is the Area.

Exam. The Base AC is 120, and the Height BD 84, to find

the Area?

B

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3) 20160 (6720 Area of the Parabola ABCD.

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21

21

6

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