Difference between revisions of "2020 AMC 10B Problems/Problem 14"
Somebody62 (talk | contribs) (→Solution: -- Could someone please edit my solution to clarify it?) |
Somebody62 (talk | contribs) (→Solution) |
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Line 72: | Line 72: | ||
J = (1,0); | J = (1,0); | ||
K = (3,0); | K = (3,0); | ||
+ | pair d = 2.7*dir(78); | ||
dot(G); | dot(G); | ||
dot(H); | dot(H); | ||
Line 80: | Line 81: | ||
label("1",(1.5,0),S); | label("1",(1.5,0),S); | ||
label("1",(2.5,0),S); | label("1",(2.5,0),S); | ||
− | + | add(pathticks(G--J,1,0.5,0,3,red)); | |
− | + | add(pathticks(I--J,1,0.5,0,3,red)); | |
− | + | add(pathticks(G--I,1,0.5,0,3,red)); | |
− | label("$60^\circ$",anglemark(H,G,I), | + | add(pathticks(G--H,1,0.5,0,3,red)); |
+ | add(pathticks(G--K,1,0.5,0,3,red)); | ||
+ | add(pathticks(K--H,1,0.5,0,3,red)); | ||
+ | label("$60^\circ$",anglemark(H,G,I),d); | ||
draw(anglemark(H,G,I,8),blue); | draw(anglemark(H,G,I,8),blue); | ||
draw(G--J--I--G); | draw(G--J--I--G); |
Revision as of 18:43, 7 February 2020
Contents
[hide]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
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 .
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 |
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