Question

A segment of height 3 meters from the center of chord to center of arc has an arc of 1/3 radians. Find the area of the segment.

Answer 1

I recall that same question where the central angle was (1/3)π radians.
I will assume you have a typo.
If not, you can still follow my method, you will simply have to change the numbers. (they will come out horrible!!)


One rotation in the circle is 2π radians or 360°
your segment forms a central angle of π/3 or 60°
So the angle formed by a radius and the chord is 60°
Ahh, an equilateral triangle

sin60° = 3/r
r = 3/sin60 = 6/√3
so the radius is 6/√3
and the length of the chord is 6/√3 , it is equilateral

area of sector = (6/√3)(6/√3)sin60 = 12(V3/2) = 6√3
area of triangle = (1/2)(6/√3)(3) = 9/√3 or 3√3 after rationalizing.

so the area of the segment = 6√3 - 3√3 = 3√3

Ahh, it looks like the chord bisects the area of the sector, when sector angle is 1/3π radians