Perimeter

Geometry Level 3

Three congruent circles are inscribed in an equilateral triangle as shown above. If each circle has a radius of $3$ . What is the perimeter of the triangle?

\frac{18}{13}

8(1+\sqrt{3}

)

9(1+\sqrt{3})

6(1+\sqrt{3}

)

18(1+\sqrt{3})

1 solution

Hana Wehbi
Jul 8, 2017

Notice that $AB$ is a side in a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle so $AB= r\sqrt{3}$ .

Then the side of the outer triangle is $2r+ (2\times r \sqrt{3}) = 6+6\sqrt{3} \implies P= 18+18\sqrt{3}$ , where $P$ is the perimeter of the outer triangle. Keep in mind $r=3$ .

Perimeter

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