TRIANGLE

 * If length  of     the        curve BEC         under blue shed is a1, then a1=2πb(360-A)/360  where A is in degree.                                                                                                    =(2π/360)*b*(360-A)=k*b*(360-A) ...(1)  and  k is a constant. k=(2π/360)                                                                                                             a1=(2π-A)*b .....(1a)              where A is in radian.                                                                                                                =f1(A)*f2(b) * If length of the curve BDC under white shed is a2, then a2= (2πb-a1) =2πb(A)/360 =k*b*A  ....(2) where A is in degree.                                                                                           a2= (2πb-a1)=  A*b  ......(2a)            where A is in radian. * Length of the chord BC is  a and                                     a=√2*b *√(1-cosA)      * a1/a  =(√2π /360)*(360-A)/ √(1-cosA) where A is in degree.   a1/a  =(2π-A)/[√2* √(1-cosA)] where A is in radian.* a2/a  =(√2π /360)*A/ √(1-cosA) where A is in degree.  a2/a  =A/[√2* √(1-cosA)] where A is in radian.*(a1+a2)/a = √2π / √(1-cosA) *  Area of the curved triangle is b2*ar/2 and area of straight triangle is BC*height/2 where BC=2bsin(A/2) and height is bcos(A/2). Hence area is A=  (b2sin(ar))/2. Therefore, Area Difference =Δarea=Curved Area-Straight area = (b2/2)*angle A* (1-[sinA/A]) . If we take function  f(b) = (b2/2) and function f''(A) = A* (1-[sinA/A]), then        Area Difference =ΔA=  f(b)*f'(A). First function is dependent on radius and second one dependent on angle A and product of both the functions contribute to area difference between the curved and the straight triangle. When A is very small close to zero radian, f(A) --> zero as sinA/A --> 1 and therefore Area Difference =Δarea --> zero which implies that the curved arc almost coincides with the straight chord. On the other hand, Δarea / area of curve BCD= = 1-(sinA/A) which implies that the ratio is only dependent on Angle and not on radius as the effect of change due to radius are equal for both numerator and denominator and cancel out. On limiting value of A--> zero, this ratio tends to zero. * On filling up values of x1,x2 and submitting, we get 6 chords [C(4,2)] connecting points , B(x1,y1a) ,B1(x1,y1b),   C(x2,y2a) & C1(x2,y2b). Value of b Angle A in Degree value of g: value of f: value of x1 value of x2 Find chord :a (BC) Find curved length a1 as per equn.(1) Find curved length a1 as per equn.(1a) Find curved length a2 as per equn. (2) Find curved length a2:(2πb-a1) as per equn. (2a) k Find (a1/a) Find (a2/a) Value of  higher limiting (a2/a) Value of  lower limiting (a2/a) (a1+a2) / a : Perimeter of st. triangle: (P) Perimeter of curved triangle:(P1) P1/P : Area of St. triangle: (A) Area of Curved triangle:(A1) A1/A : a1/2b: Find   y1: and Find   y2: and Find chord BB1: Find chord CC1: Find chord B1C: Find chord BC1: Find chord B1C1: Area of white space BCD: Δarea =(b2/2)(angle BAC- sin angleBAC) Δarea / area ABDC (in%) = [1-(sin of angle BAC/angle BAC)]*100