# CAT Practice : Mensuration

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What is the sum of interior angles in a pentagon ?

## Pentagon and Circles

Q.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. ${\pi}$r2
2. 1.5${\pi}$r2
3. 2${\pi}$r2
4. 1.25${\pi}$r2

Choice B. 1.5${\pi}$r2

## Detailed Solution

Diagram -

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 ${\frac{\pi*r^2}{360°}}$

In this case,

Sum of shaded region = ${\angle{P} * \frac{\pi*r^2}{360°}}$ + ${\angle{Q} * \frac{\pi*r^2}{360°}}$ ………… + ${\angle{T} * \frac{\pi*r^2}{360°}}$

= (P + Q +…. T) x ${\frac{\pi*r^2}{360°}}$

= Sum of interior angles of pentagon x ${\frac{\pi*r^2}{360°}}$

= 3 x 180° x ${\frac{\pi*r^2}{360°}}$(Sum of interior angles = (n – 2) X 180°

= 3 * ${\frac{\pi*r^2}{2}}$

Note => The above concept is applicable for a polygon of n sides.

Choice (B) is therefore, the correct answer.

Correct Answer: 1.5${\pi}$r2

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## More questions from Mensuration

Mensuration is the science of measurement. Cylinders, cones, cuboids, rectangular parallelopipeds etc.