Difference between revisions of "1958 AHSME Problems/Problem 16"
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== Solution == | == Solution == | ||
− | <math>\ | + | We can split the hexagon into 6 equilateral triangles. If the area of the circle is <math>100pi</math>, then the radius is <math>10</math>. The radius is equal to the height of one of the equilateral triangles. Using <math>sin60 = \frac{\sqrt3}{2}</math>, we get the hypotenuse is <math>\frac{20\sqrt{3}}{3}</math>, which is also equal to the side length. Using the hexagon formula (or going back to the equilateral triangle formula <math>\cdot 6</math>), we get |
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+ | <math>\frac{6}{4} \cdot s^2\sqrt{3} \Rightarrow</math> <math>\frac{6}{4} \cdot (\frac{20\sqrt3}{3})^2 \cdot \sqrt3 \Rightarrow</math> <math> \frac{6}{4} \cdot \frac{20\sqrt3}{3} \cdot \frac{20\sqrt3}{3} \cdot \sqrt3 \Rightarrow</math> <math>\boxed{200\sqrt3}</math> | ||
== See Also == | == See Also == |
Latest revision as of 01:38, 12 March 2017
Problem
The area of a circle inscribed in a regular hexagon is . The area of hexagon is:
Solution
We can split the hexagon into 6 equilateral triangles. If the area of the circle is , then the radius is
. The radius is equal to the height of one of the equilateral triangles. Using
, we get the hypotenuse is
, which is also equal to the side length. Using the hexagon formula (or going back to the equilateral triangle formula
), we get
See Also
1958 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AHSME Problems and Solutions |
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