Let side of regular hexagon be a.
The shortest diagonal will be of length a3. Why?
A regular hexagon is just 6 equilateral triangles around a point. The shortest diagonal is FD.
FD = FP + PD
FOE is equilateral and so is EOD.
Diagonal FD can be broken as FP + PD, both of which are altitude of equilateral s.
FP = (3a)/2
FD = 3 a = shortest diagonal
The question tells us that the shortest diagonal measures 3 cm.
3 a = 3 => a = 3
Radius of circle = 3
Area of hexagon = (√3 a ^{2})/4 x 6
Area of circle – area of hexagon = π (√3)2 − √3/4 x (√3)2 x 6
= 3π − (9√3)/2
Area of shaded region = 1/(6 ) (area(circle) – area(hexagon))
= 1/(6 )(3π − (9√3)/2)
Answer choice (a).
Correct Answer: 1/6(3 − (93)/2)