2021 Fall AMC 10B Problems/Problem 11
Contents
Problem
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 (Best)
This is the graph of the original Hexagon. After reflecting each minor arc over the sides of the hexagon it will look like this;
This bounded region is the same as the area of the hexagon minus the area of each of the reflect arcs. From the first diagram, the area of each arc is the area of the sector minus the area of the equilateral triangle, so each arc has an area of .
There are 6 total arcs, so the total area of the arcs is .
The area of the hexagon is , so the area of the bounded region is:
~KingRavi
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
Solution 3
Denote by the center of this circle. Hence, the radius of this circle is 1. Denote this hexagon as .
We have . Hence, the area of the region formed between segment and the minor arc formed by and , denoted as , is
Therefore, the area of the region that this problem asks us to compute is
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 4
The area of the desired shape is equal to the whole hexagonSix arcs
Since the arcs are reflected upon the sides of the hexagon, we can see that Six arcsWhole circleWhole hexagon.
The area of the circle is , since the shape has side length and is inscribed within the circle (so its diameter is ).
Combining these two, we see that From here, three solutions can progress:
Solution 4.1 (General polygons)
The area of a regular polygon is equal to , where is the perimiter of the polygon, and is the length of its apothem.
The apothem is the distance from the center of the polygon to the midpoint of two adjacent vertices. If we were to create an equilateral triangle, whose base is at the side of the polygon, and its two sides meeting at the center, its height would be the apothem.
In this case, the side length of the hexagon is . We can now split this equlilateral into two congruent right triangles, who both have the apothem as a side. Each triangle has side lengths of ,half of the base, , the radius, and the apothem. By the pythagorean theorem, , for the apothem . Solving yields
Since , our formula becomes
Reall that the area of the figure is so the sought area is , so our answer is
Alternatively, we could find the length of the apothem by the formula where is the side length and is the number of sides.
Solution 4.2 (Quick)
The area of a regular hexagon is given by the formula , for the side length , which in this case, is , so the area of this hexagon is
We seek which is
~Benedict T (countmath1)
Video Solution by Interstigation
Video Solution
~Education, the Study of Everything
Video Solution by WhyMath
~savannahsolver
Video Solution by TheBeautyofMath
~IceMatrix
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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