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 $?
+2862 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
