ABCDEF is a regular hexagon inscribed inside a circle. If the shortest diagonal of the hexagon is of length 3 units, what is the area of the shaded region?

- 1/6(3π − (9√3)/2)
- 1/6(2π − (6√3)/2)
- 1/6(3π − (8√3)/2)
- 1/6(6π − (15√3)/2)

Let side of regular hexagon be a.

The shortest diagonal will be of length a√3. 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 = √3a = shortest diagonal

The question tells us that the shortest diagonal measures 3 cm.

√3a = 3 => a = √3

Radius of circle = √3

Area of hexagon = (√3a^{2} )/4 * 6

Area of circle – area of hexagon = π (√3)2 − √3/4 * (√3)2 * 6

= 3π − (9√3)/2

Area of shaded region = 1/(6 ) (area(circle) – area(hexagon))

= 1/(6 )(3π − (9√3)/2)

The question is **"what is the area of the shaded region?"**