BM 



MATHEMATICS. TRIGONOMETRY. 



[ARKAS or 



Similarly 



( 



(43). To find the Area of the Circumscribed Circle. 



Let A B C bo the triangle. 

 Circumscribe a circle about 

 it, the centre of which is O. 

 Join B O and produce it to 

 meet the circumference in 

 D. Join DC. Then (Eu- 

 clid III. 21) B D C = 

 BAC, and BCD is a right 

 angle. (Euclid III. 31.) 

 Now B D sin. B D C = 

 BC. 



.'. If R = radius of cir- 

 cumscribed circle, 2 R. sin. 



Area of triangle A B C = 



I 



AB, BO . 

 g sin. BC. 



/. The whole 

 Hence area of quadrilateral 



o)( b) ( e) 



abc 



R = 



(44). To find the Area of a Quadrttatei-al inscribed in a 



Circle in terms of the side*. 



A B C D a quadrilateral inscriptible in a circle, let the 

 four sides A B, BC, CD, DA, be 

 respectively abed. Join A C. 



Now, ifABC = ADC = 

 180 (Euclid III. 22.) 

 Hence, A C" = a 2 + b* 2a6 



cos. 

 and A C 2 = c a +d* Zed cos. 



(180 0) 



.-. o + b* 2a6 cos. 6 = c s + d* 

 + 2cd cos. 



.'. o 5 + W Zdb cos. 6 = c 2 + <2 2 + 2cd cos. 



a> + W c d* 

 .-.cos.0= 



Now, 2coB.*y ' 

 



.'. 2 COS. 1 77 



1 + COS. 



1 + -~ L 



2 - o- 



Similarly, 



2in.'|-l- 2o6 + 2c<r 



ab+cd 



(ab+cd? 

 Bat nit a of triangle ADO 



(45). To find the Area of a Polygon of (n) sides inscribed 

 in a given Circle. 



If A B be one side of the polygon, O the centre of ; 

 Fig. 21. the circle, let r be the radius, and 



71 the number of sides. Then area , 

 of polygon = n X (area triangle 

 OAB). 



Draw Op perpendicular to A B, 

 Then 



JQAO 



AOp= JAOB .'. ' 



Then area of triangle 



r sin. L. Op. 



r cos. 



180 



'. Area triangle = r 2 sin. cos. , 



n n 



J r 3 sin. 



300 



If a be one side of the polygon, 



Then ^=rsin. ^21. 

 2 n 



180 

 2sm. 



.'. area of triangle : 



Hence area of polygon ^r 3 sin. when radius is 

 & n 



given, 

 And area of polygon ^-cotan. - when side ia 



given. 



The area of circumscribed polygon can in like manner 

 be proved to equal 



nr 1 tan. 



ISO" 



