Difference between revisions of "2021 Fall AMC 10B Problems/Problem 11"
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bounded by these <math>6</math> reflected arcs? | bounded by these <math>6</math> reflected arcs? | ||
| − | ==Solution== | + | ==Solution 1== |
| + | |||
| + | <asy> | ||
| + | import olympiad; | ||
| + | unitsize(50); | ||
| + | Pair A,B,C,D,E,F,O; | ||
| + | A = origin; B = (-0.5,0.866025); C=(0,1.7320508); D=(1,1.7320508); E=(1.5,0.866025); F=(1,0); | ||
| + | draw(A--B--C--D--E--F--cycle); | ||
| + | |||
| + | |||
| + | ==Solution 2== | ||
Let the hexagon described be of area <math>H</math> and let the circle's area be <math>C</math>. | Let the hexagon described be of area <math>H</math> and let the circle's area be <math>C</math>. | ||
Let the area we want to aim for be <math>A</math>. | Let the area we want to aim for be <math>A</math>. | ||
Revision as of 14:19, 24 November 2021
Contents
Problem 11
A regular hexagon of side length 
is inscribed in a circle. Each minor arc of the circle
determined by a side of the hexagon is reflected over that side. What is the area of the region
bounded by these 
reflected arcs?
Solution 1
<asy> import olympiad; unitsize(50); Pair A,B,C,D,E,F,O; A = origin; B = (-0.5,0.866025); C=(0,1.7320508); D=(1,1.7320508); E=(1.5,0.866025); F=(1,0); draw(A--B--C--D--E--F--cycle);
Solution 2
Let the hexagon described be of area 
and let the circle's area be 
.
Let the area we want to aim for be 
.
Thus, we have that 
, or 
.
By some formulas, 
and 
.
Thus, 
or 
.
~Hefei417, or 陆畅 Sunny from China
See Also
| 2021 Fall AMC 10B (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 10 Problems and Solutions | ||
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