# CAT Practice : Geometry-Triangles

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Hexagon inscribed inside circle

## Hexagon inscribed inside circle

Q.13: 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. 1/6(3${\Pi }$ − (9${\surd }$3)/2)
2. 1/6(2${\Pi }$ − (6${\surd }$3)/2)
3. 1/6(3${\Pi }$ − (8${\surd }$3)/2)
4. 1/6(6${\Pi }$ − (15${\surd }$3)/2)

Choice (A). 1/6(3${\Pi }$ − (9${\surd }$3)/2)

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## Detailed Solution

Let side of regular hexagon be a.

The shortest diagonal will be of length a${\surd }$3. Why?

A regular hexagon is just 6 equilateral triangles around a point. The shortest diagonal is FD.

FD = FP + PD

${\triangle }$FOE is equilateral and so is ${\triangle }$ EOD.

Diagonal FD can be broken as FP + PD, both of which are altitude of equilateral s.

FP = (${\surd }$3a)/2

FD = ${\surd }$3 a = shortest diagonal

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

${\surd }$3 a = 3 => a = ${\surd }$3

Radius of circle = ${\surd }$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)

Correct Answer: 1/6(3${\Pi }$ − (9${\surd }$3)/2)

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Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.