Difference between revisions of "2020 AMC 10B Problems/Problem 14"
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<math> \textbf {(A) } 6\sqrt{3}-3\pi \qquad \textbf {(B) } \frac{9\sqrt{3}}{2} - 2\pi\ \qquad \textbf {(C) } \frac{3\sqrt{3}}{2} - \frac{\pi}{3} \qquad \textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math> | <math> \textbf {(A) } 6\sqrt{3}-3\pi \qquad \textbf {(B) } \frac{9\sqrt{3}}{2} - 2\pi\ \qquad \textbf {(C) } \frac{3\sqrt{3}}{2} - \frac{\pi}{3} \qquad \textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math> | ||
− | ==Solution== | + | ==Solution 1== |
<asy> | <asy> | ||
real x=sqrt(3); | real x=sqrt(3); |
Revision as of 18:25, 9 February 2020
Problem
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region — inside the hexagon but outside all of the semicircles?
Solution 1
Let point A be a vertex of the regular hexagon, let point B be the midpoint of the line connecting point A and a neighboring vertex, and let point C be the second intersection of the two semicircles that pass through point A. Then, , since B is the center of the semicircle with radius 1 that C lies on,
, since B is the center of the semicircle with radius 1 that A lies on, and
, as a regular hexagon has angles of 120
, and
is half of any angle in this hexagon. Now, using the sine law,
, so
. Since the angles in a triangle sum to 180
,
is also 60
. Therefore,
is an equilateral triangle with side lengths of 1.
Since the area of a regular hexagon can be found with the formula , where
is the side length of the hexagon, the area of this hexagon is
. Since the area of an equilateral triangle can be found with the formula
, where
is the side length of the equilateral triangle, the area of an equilateral triangle with side lengths of 1 is
. Since the area of a circle can be found with the formula
, the area of a sixth of a circle with radius 1 is
. In each sixth of the hexagon, there are two equilateral triangles colored white, each with an area of
, and one sixth of a circle with radius 1 colored white, with an area of
. The rest of the sixth is colored gray. Therefore, the total area that is colored white in each sixth of the hexagon is
, which equals
, and the total area colored white is
, which equals
. Since the area colored gray equals the total area of the hexagon minus the area colored white, the area colored gray is
, which equals
.
Solution 2
First, subdivide the hexagon into 24 equilateral triangles with side length 1:
Now note that the entire shaded region is just 6 times this part:
The entire rhombus is just 2 equilatrial triangles with side lengths of 1, so it has an area of:
The arc that is not included has an area of:
Hence, the area of the shaded region in that section is
For a final area of:
~N828335
Video Solution
~IceMatrix
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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