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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

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