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# [Smart Math] Geometry Problem 12

Here’s and example of a SMART MATH problem for GEOMETRY. ### Problem

Find the areas of the shaded region in the equilateral triangle. 1. $\frac{\pi }{a^{2}}$sq. units
2. $3a^{2}-a$sq. units
3. $\frac{\pi }{3}\left( a^{2}-1 \right)$sq. units
4. $\frac{a^{2}}{4}\left( \sqrt{3}-\pi \right)$sq. units
5. $a\left( \frac{\pi }{3}-1 \right)$sq.units

### The Usual Method

Area of the equilateral triangle = $\frac{\sqrt{3}}{4}a^{2}$sq. units.

Area of each part of the circle = $\frac{1}{3}\pi \left( \frac{a}{2} \right)^{2}$

Hence area of the shaded portion = $\frac{\sqrt{3}}{4}a^{2}-3\times \frac{1}{3}\pi \left( \frac{a}{2} \right)^{2}$

= $\frac{\sqrt{3}a^{2}}{4}-\frac{\pi a^{2}}{4}=\frac{\sqrt{3}a^{2}-\pi a^{2}}{4}$

= $\frac{a^{2}}{4}\left( \sqrt{3}-\pi \right)$sq. units

(Ans: 4)

Estimated Time to arrive at the answer = 30 seconds.

### Using Technique

Since there are three equal parts of a circle in the equilateral diagram, the above diagram is same as the one shown below as we can take three parts of the circle as one complete circle. Hence area of the shaded region = Area of triangle – Area of circle. $\therefore \frac{\sqrt{3}a^{2}}{4}-\frac{\pi a^{2}}{4}$ = $\frac{a^{2}}{4}\left( \sqrt{3}-\pi \right)$sq. units

(Ans: 4)

Estimated Time to arrive at the answer = 15 seconds.
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