Six congruent circles are packed into an equilateral

[swerve]

Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

$$1. \text{ The radius of each circle is } 2.$$
$$2. \text{ The area of the triangle is } 48 + 28\sqrt{3}$$

The OA is D

Source: Veritas Prep

[fskilnik@GMATH]

Six congruent circles are packed into an equilateral triangle so that no circle is overlapping and such that circles are tangent to one another or the triangle at any point of contact, as shown above. What is the area of the part of the triangle that is NOT covered by circles?

$$1. \text{ The radius of each circle is } 2.$$
$$2. \text{ The area of the triangle is } 48 + 28\sqrt{3}$$
Source: Veritas Prep

$$?\,\,\, = \,\,\,{S_\Delta } - 6 \cdot {S_ \circ }$$
The GMATH widget (a.k.a "GW") is defined in the figure below. It is formed combining an equilateral triangle (side length 4r) and three "legs" (length r each), where r is any positive number.



With the GW in mind, the answer is (D) immediately:

(1) The value of r is given, hence the corresponding GW is unique, hence the triangle that circumscribes the GW is unique (dotted triangle in the figure shown in the bottom-left). Our FOCUS is unique.

(2) The equilateral triangle is given, hence the corresponding inscribed GW is unique (dotted GW shown in the bottom-right), hence r is unique. Our FOCUS is unique.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.