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Consider a circle with radius 1 and center $O$. Points $A, B $ are on the circumference such that $ \angle AOB = \frac{ \pi}{3} $. 

What is the radius of the largest circle that can be inscribed in sector $AOB $?

 Oct 26, 2019

A circle has 2 radians


Since AOB is \(\frac{\pi}{3}\)


AOB is 60 degrees.


AOB forms an equilateral triangle.


Find the  largest circle that fits in a equilateral triangle with sides of 1.


The formula -> \(A=rs\)


S is semi pereimeter while the R means radius.


So we solve for R.   \(r=\frac{A}{s}\)


Semi perimeter is 1 + 1 + 1 = 3


3/2 = 1.5


The area of an equilateral triangle is \(\frac{\sqrt{3}}{4}a^2\), where a is the length of one side.


So the area is \(\frac{\sqrt{3}}{4}\).


Ok now we evaluate. \(\frac{\frac{\sqrt{3}}{4}}{1.5}\).


\(\frac{\sqrt{3}}{6}\) that is the answer I think

 Oct 26, 2019

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