M ENS U RAT I ON. 



55 



From C, the centre of the inscribed circle, draw CD 

 perpendicular to the -.Me AB, and join CA, CB. Then 

 CD will be the ruliu* of the circle, and AB will be bi. 

 sected in D, al*> the angle* at A and B will be bi- 

 eected by AC nd BC. 



In the right angled triaagJe ADC rad. : tan. A : : 

 trigonometry) ; hence the truth of the first 

 part of the 



Again, since the polygon is equal to the triangle ACB 



taken a* often a* the figure ha* sides, and thi* apace i* 



manifestly equal to a rectangle contained by CD and 



AB taken a* often a* the figure ha* tide*, that i*. to 



D and half the perimeter of the figure, therefore the 



area of the polygon it equal to a rectangle contained by 



CD and hafr its perimeter. 



Ex. Find the area at a hexagon, the aide being SO 

 yard*. In thai caae, half the angle of the polygon i* 

 60*. 



Rad 1 0.0 000 



Tan 60* 102X856 



Half the tide, 10 1.00000 



of rad. of in*, circle 

 Half per. 60 



I.-MS.V, 



1.77815 

 3.01671 



PaoBiiv \'\ 



To 



KILC. 



o find the area of any rectilineal figure. 



Rnolve the figure into triangle* and trape. 

 S and compute their areas separately ; the turn will 

 be ana of the rectilineal figure a* M aaakieatly evU 



of A CJTCM, 



PaoaLtvi VII 



To find taW caMmAcr AD 

 the ane from the other. 



RULE I. A 7 to tt, to M the diameter to the or- 

 cnmference nearly. 



A* SS u to 7, to i* the circuarfocnee to the diame- 



Ci nr...-i\. 



RULE 9. A* I IS U to 355, to U the diameter to the 



irranffrrncr nearly. 

 A* 3*5 is to 1 13, to is the circumference to the dia- 



by 3.1416, the pro- 



aaatar nearly. 



Row 8. Multiply the 

 duet i. the dmnfcm* newly. 



Divide the circumference by 3.1416, the quotient i* 

 tke diameter nearly. 



The truth of these rate* will eppear from Prop. 10. 

 of Sect. V. GEOMETRY. 



Ex. I. The diameter of a circle if 12 feet, what i* it* 



By prob. 7. .3.1416 time* the radius U half the cir- 

 curaference. which i* an arc of 180 ; and arcs of a cir- 

 cle have the came ratio a* their measure* expressed in 

 degree*. 



Ex. Reauired the length of the arc ADB, whose 

 chord AB it 6, the radius being 9 ? 



From C, thie centre, draw CD, bitecting the arc ; 

 this line will be perpendicular to the chord, and will 

 bisect it. (Gt. 6. ^). 



By Trigonometn-. CA = 9 Ar. Comp. 9.04576 

 i*toAP=3 0.47718 



a* rad. 



Again, 



To tin. ACP = 1 9* 28f 9-52288 

 Hence ACB a> JT fflf' 38.944 



180 : 38.944 : : 3.1416 X 9 : ****** = 



ISO 



IU-LE II. A near approximation to any arc of a circle 

 may be found by thi* proportion, 



9 rao*. -f 6 cot. a : 14 raa*. + CM. a : : $in.a: arc a. 

 This approximation wa investigated a* follow*. 

 ftipyeaiug the radio* of the circle to be unity, let as 



A in. a -f. B tin. 2 a = a (C + D ro*. a), 

 in which A, B, C, D are number* to be presently de- 

 termined, and a* we arc seeking only an approximation 

 to the are, let u* lappoee it to be tuch a fraction of the 

 radiu* that it* 7th an>l higher power* may be neglect- 

 ad. Then by well known exprmioni for the tine and 

 ooaiat of an arc. Sea AMITUMETIC o SIMM, { 29, 



' a' 



S,n. .= ,-_ + __ 



8 a* 39a' 

 ' 6- + T20 



IT- 



a Co*, a = a 



Let these value* of sin. a, tin. 2 a, *nd a cot. a, be 

 substituted in the MSBIIIIIJ equation, and then, by ma- 

 king the co-efficient* of like power* of a equal to each 

 other, we find 



: (' + D, A + 8B=3D, A -f 326=5 D; 

 and hence A ** 



Bjr raW let, 7 : : : If: 



By rale 3d. 3.1416X 1'2=S7.699, 



By rale 2d, 113:355:: : - = 37-699115 



I I J 



3 ' 12^ 2 ' 



Tbe*e v*)ue of A, B. C being substituted in the a*. 

 anted equation, and 2 Sin. a Co*, a put instead of 

 Sin. Sa, we hae 



Sin. (14 + cot. a) -t (9 + 6co*.o). 

 .71 feet, nearly. Hence the truth of the rule is evident. 



Taking the taioe example a* ia last problem, we fcave 

 rad.- 9, and ain.J as 3; hence cor 4 a =. ^/ (rad.* 

 am.' 4 a) = </~i = 8.4*3528, and we have by the rule 



13,.9,.68:,34.485M::8: IM4Mi8 *- 



rtoth. truth 



Ex t. The ornitnfrrrnrf of a circle M one pole, or *j 

 yard*, what i* it* diameter? 



: 1.757 yard*. The diameter. 



POOBLXM VIII. 

 T* find the length of any arc of a circle. 



<l the number of degree* in the arc; theti 

 a* \*f i* to that number, n i* 3.1416 time* the radjin 

 to the length ot the arc. 



131.91168 



Hence the length of the arc i* 6.1 1706 ; thi* value is 

 true to the last figure. 



NOTE. For another approximate value to an arc of a 

 circle, tee GEOMETRY, Prep> II. Sect. 5. Part 1. 



PaoaLEM IX. 



To find the area of a circle. 



RULE I. Multiply the radiu* by half the circumfe- 

 rence, the product it the i 



