CAT Practice : Geometry-Triangles

You are here: Home  CAT Questionbank   CAT Quant  Geometry: Triangles  Question 13
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 − (93)/2)
    2. 1/6(2 − (63)/2)
    3. 1/6(3 − (83)/2)
    4. 1/6(6 − (153)/2)

 

  • Correct Answer
    Choice (A). 1/6(3 − (93)/2)

Explanatory Answer

Click to watch video solution
Click to view the explanation as a slide show

Detailed Solution



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)



Our Online Course, Now on Google Playstore!

2IIM's App

Fully Functional Course on Mobile

All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.


Cache Content for Offline Viewing

Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!

Get it on Google Play
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.