Problem 172

{{searchResults.length}} results for “{{searchWords}}” No results found for “{{searchWords}}” × back to contents

§{{x[0]}} {{x[1]}}

Table of Contents

Problem 172: (2019 AMC 10a 16; 2019 AMC12a 10)

The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$

$\textbf{(A) } 4 \pi \sqrt{3} \qquad$ $\textbf{(B) } 7 \pi \qquad$ $\textbf{(C) } \pi\left(3\sqrt{3} +2\right) \qquad$ $\textbf{(D) } 10 \pi \left(\sqrt{3} - 1\right) \qquad$ $\textbf{(E) } \pi\left(\sqrt{3} + 6\right)$

Solution

Exercise 172: Solution

The figure shows $7$ circles of radius $2$ within a larger circle: one at the center and six around it, each tangent to the center circle and to its two neighbors, with all outer circles tangent to the large circle. All intersections occur at points of tangency. What is the area of the region inside the larger circle but outside all $7$ circles of radius $2$?

Search Go to Prev Next

Contact Us