# CAT Practice : Coordinate Geometry

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Very simple Question!! Go about it by focussing on the radius of the circle or side of the Hexagon.

## Ratio of Areas

Q.28: Consider Regular Hexagon H inscribed in circle C, what is ratio of the areas of H and C? Consider Circle C inscribed in Regular Hexagon H, what is ratio of the areas of H and C?
1. 2${\surd 3}$ : ${3\Pi}$ , 3${\surd 3}$ : ${4\Pi}$
2. 3${\surd 3}$ : ${\Pi}$ , 3${\surd 3}$ : ${4\Pi}$
3. 3${\surd 3}$ : ${2\Pi}$, 2${\surd 3}$ : ${\Pi}$
4. ${\surd 3}$ : ${\Pi}$ , ${\surd 3}$ : ${4\Pi}$

Choice (C). 3${\surd 3}$ : ${2\Pi}$, 2${\surd 3}$ : ${\Pi}$

## Detailed Solution

Consider regular hexagon of side ‘a’.

Inradius = altitude of equilateral triangle of side a = ${\frac {√3a}{2} }$

Area of regular hexagon = 6 * ${\frac {√3a^2}{4} }$

Area of smaller circle = ${\pi * \frac {√3a}{2} * \frac {√3a}{2}}$ = ${\frac {3\pi a^2}{4} }$

Area of larger circle = ${\pi a^2 }$

When Hexagon H inscribed in circle C,
Ratio of Areas of H and C :: 3${\surd 3}$ : ${2\Pi}$

When Circle C inscribed in Regular Hexagon H,
Ratio of Areas of H and C :: 2${\surd 3}$ : ${\Pi}$

Correct Answer: (C). 3${\surd 3}$ : ${2\Pi}$, 2${\surd 3}$ : ${\Pi}$

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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.