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The question is from regular polygon. It discusses about a regular hexagon inscribed inside a circle. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems. Triangles are heavily tested, the wonderful infinite-sided polygon that is the circle is also heavily tested. In between these two lies this great mass of regular polygons.

Question 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.
CAT Question - Geometry - Regular Octagon

  1. 1/6(3ฯ€ โˆ’ (9โˆš3)/2)
  2. 1/6(2ฯ€ โˆ’ (6โˆš3)/2)
  3. 1/6(3ฯ€ โˆ’ (8โˆš3)/2)
  4. 1/6(6ฯ€ โˆ’ (15โˆš3)/2)


Video Explanation


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Explanatory Answer

Method of solving this CAT Question from Regular Hexagon: A regular hexagon is just 6 equilateral triangles around a point.
CAT Question - Geometry - Regular Octagon

Let side of regular hexagon be a.
The shortest diagonal will be of length aโˆš3. Why?
A regular hexagon is just 6 equilateral triangles around a point. The shortest diagonal is FD.
FD = FP + PD
โ–ณFOE is equilateral and so is โ–ณ EOD.
Diagonal FD can be broken as FP + PD, both of which are altitude of equilateral ๏?„s.
FP = (โˆš3a)/2
FD = โˆš3a = shortest diagonal
The question tells us that the shortest diagonal measures 3 cm.
โˆš3a = 3 => a = โˆš3
Radius of circle = โˆš3
Area of hexagon = (โˆš3a2 )/4 * 6
Area of circle โ€“ area of hexagon = ฯ€ (โˆš3)2 โˆ’ โˆš3/4 * (โˆš3)2 * 6
= 3ฯ€ โˆ’ (9โˆš3)/2
Area of shaded region = 1/(6 ) (area(circle) โ€“ area(hexagon))
= 1/(6 )(3ฯ€ โˆ’ (9โˆš3)/2)

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

Hence, the answer is 1/(6 )(3ฯ€ โˆ’ (9โˆš3)/2)

Choice A is the correct answer.

ย 


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