Question

Express answer in exact form.

Answer 1

We know that
the regular hexagon can be divided into 6 equilateral triangles

area of one equilateral triangle=s²*√3/4
for s=3 in
area of one equilateral triangle=9*√3/4 in²

area of a circle=pi*r²

in this problem the radius is equal to the side of a regular hexagon
r=3 in
area of the circle=pi*3²-----> 9*pi in²
we divide that area into 6 equal parts------> 9*pi/6----> 3*pi/2 in²

the area of a segment formed by a side of the hexagon and the circle is equal to 1/6 of the area of ​​the circle minus the area of ​​1 equilateral triangle
so
 [ (3/2)*pi in²-(9/4)*√3 in²]

the answer is
 [ (3/2)*pi in²-(9/4)*√3 in²]