# CAT Quantitative Aptitude Questions | CAT Geometry - Mensuration Problems

###### CAT Questions | CAT Mensuration Questions | Pentagon and Circles
CAT Questions

This question is from CAT Mensuration in CAT Geometry.In this question, five circles are inscibed in each of the vertices of a pentagon. And we need to find out the overlapping area of the figures. Mensuration Questions in the CAT exam can be solved with practical approach and thought process rather than just knowing theorems from CAT Geometry. Even if one is not completely prepared for CAT Geometry exam, he or she should be able to do well in Mensuration problems tested by the CAT Exam. Knowing Mensuration formulas is important. Make sure you master Mensuration questions for CAT exam.

Question 14: PQRST is a pentagon in which all the interior angles are unequal. A circle of radius ‘r’ is inscribed in each of the vertices. Find the area of portion of circles falling inside the pentagon.

1. πr2
2. 1.5πr2
3. 2πr2
4. 1.25πr2

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##### Method of solving this CAT Question from Geometry - Mensuration: What is the sum of interior angles in a pentagon ?

Since neither angles nor sides are given in the question, immediately the sum of angles of pentagon should come in mind. To use it,
We know the area of the sectors of a circle is given as,
Central Angle x π∗r2/360°
In this case,
Sum of shaded region = ∠P∗π∗r2/360° + ∠Q∗π∗r2/360° ………… + ∠T∗π∗r2/360°
= (P + Q +…. T) x π∗r2/360°
= Sum of interior angles of pentagon x π∗r2/360°
= 3 x 180° x π∗r2/360°(Sum of interior angles = (n – 2) X 180°)
= 3 * π∗r2/2
Note => The above concept is applicable for a polygon of n sides.

The question is "Find the area of portion of circles falling inside the pentagon."

##### Hence, the answer is 1.5πr2

Choice B is the correct answer.

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