A regular pentagon is inscribed in a circle. find the area of sector which each side of the pentagon subtends at the centre
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Answer:
Step-by-step explanation:
Area of n-sided regular Polygon =
2
l
2
n
sin(
n
2π
)=k
2
ntan(
n
π
),
where l is the length of the half of it's diagonal,
k is the length of the half of the perpendicular bisector from one side to it's opposite side (k=lcos(
n
π
))
and n is the no of sides of the polygon.
Here, l
I
n
=k
O
n
= radius of circle=1
So, I
n
=
2
n
sin(
n
2π
)
And, O
n
=ntan(
n
π
)
ie, I
n
=
2
O
n
2cos
2
(
n
π
)=
2
O
n
(1+cos(
n
2π
))=
2
O
n
(1+
1−sin
2
(
n
2π
)
)
⇒I
n
=
2
O
n
⎝
⎛
1+
1−(
n
2I
n
)
2
⎠
⎞
Q.E.D
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