Question
Tue March 03, 2015 By: Gourav Barik

Suppose in a circle of centre O, the radius are joined with the end points of a chord, A and B. Now we have to find the minor segment subtended by the chord. So, we have 
Area of sector - Area of triangle = Area of segment
Now, how to find the area of triangle if it's not an equilateral triangle? Take thita anything besides 60 degree and length of chord and radii from your choice.
Thank you!

Expert Reply
Vimala Ramamurthy
Thu March 05, 2015
C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
H e r e comma space A r e a space o f space A M B C equals A r e a space o f space t h e space s e c t o r space O A C B minus A r e a space o f space t r i a n g l e space O A B C o n s i d e r space t h e space c i r c l e space o f space r a d i u s comma space 4 square root of 2 space c m comma space a n d space l e n g t h space o f space c h o r d space 8 space c m T h a t comma space O A equals O B equals 4 square root of 2 space c m space a n d space A B equals 8 space c m S i n c e space t h e space p e r p e n d i c u l a r space l i n e space j o i n i n g space t h e space c e n t r e space o f space t h e space c i r c l e space a n d space t h e space c h o r d space b i s e c t s space t h e space c h o r d comma space w e space h a v e comma space A M equals M B equals 4 space c m N o w space c o n s i d e r space t h e space r i g h t space t r i a n g l e B y space a p p l y i n g space P y t h a g o r a s space t h e o r e m comma space w e space h a v e comma O A squared equals A M squared plus O M squared rightwards double arrow O M squared equals O A squared minus A M squared rightwards double arrow O M squared equals open parentheses 4 square root of 2 close parentheses squared minus 4 squared rightwards double arrow O M squared equals 32 minus 16 rightwards double arrow O M squared equals 16 rightwards double arrow O M equals 4 T h u s comma space t h e space a r e a space o f space t h e space t r i a n g l e space O A B equals 1 half cross times A B cross times O M equals 1 half cross times 8 cross times 4 equals 16 space s q. space u n i t s N o w space a r e a space o f space t h e space s e c t o r space w i t h space a n g l e space theta equals 90 degree comma space t h e n space A r e a space o f space O A C B equals fraction numerator theta over denominator 360 degree end fraction cross times straight pi cross times straight r squared rightwards double arrow straight A straight r straight e straight a space straight o straight f space straight O straight A straight C straight B equals fraction numerator 90 degree over denominator 360 degree end fraction cross times straight pi cross times open parentheses 4 square root of 2 close parentheses squared rightwards double arrow straight A straight r straight e straight a space straight o straight f space straight O straight A straight C straight B equals 1 fourth cross times straight pi cross times 16 cross times 2 equals 8 straight pi space sq. units straight A straight r straight e straight a space straight o straight f space straight A straight M straight B straight C equals straight A straight r straight e straight a space straight o straight f space straight t straight h straight e space straight s straight e straight c straight t straight o straight r space straight O straight A straight C straight B minus straight A straight r straight e straight a space straight o straight f space straight t straight r straight i straight a straight n straight g straight l straight e space straight O straight A straight B rightwards double arrow straight A straight r straight e straight a space straight o straight f space straight A straight M straight B straight C equals 8 straight pi minus 16
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