2020 AMC 12B Problems/Problem 11
Problem
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 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
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
See Also
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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