Difference between revisions of "2023 AMC 10A Problems/Problem 24"
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Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is <math>\frac{3}{7}</math> unit. What is the area of the region inside the frame not occupied by the blocks? | Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is <math>\frac{3}{7}</math> unit. What is the area of the region inside the frame not occupied by the blocks? | ||
| − | + | [asy] | |
unitsize(1cm); | unitsize(1cm); | ||
draw(scale(3)*polygon(6)); | draw(scale(3)*polygon(6)); | ||
| Line 10: | Line 10: | ||
filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray); | filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray); | ||
filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray); | filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray); | ||
| − | [/asy] | + | [/asy] |
<math>\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}</math> | <math>\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}</math> | ||
Revision as of 16:30, 9 November 2023
Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is 
unit. What is the area of the region inside the frame not occupied by the blocks?
[asy]
unitsize(1cm);
draw(scale(3)*polygon(6));
filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray);
filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray);
[/asy]
