document.write( "Question 772850: A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters; how do I find the radius of the circle, the length of the side of the hexagon, and the perimeter of the hexagon. \n" ); document.write( "

Algebra.Com's Answer #471253 by Edwin McCravy(20055) $\"\"$ $\"About$

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document.write( "The apothem divides the bottom side into two equal parts. I will\r\n" );
document.write( "label each part x. Then the whole bottom side of the hexagon will \r\n" );
document.write( "be 2x.  We want to find x so we can find the bottom side 2x.\r\n" );
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document.write( "Draw lines to the vertices\r\n" );
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document.write( "Each of the central angles is 60° because  = 60°.\r\n" );
document.write( "So all the triangles formed are equilateral, because they are isosceles\r\n" );
document.write( "because the sides are radii of the same circle, and an isosceles\r\n" );
document.write( "triangle with one 60° angle is equilateral.  Since all three\r\n" );
document.write( "sides of an equilateral triangle have the same measure, the bottom \r\n" );
document.write( "side is 2x, and so are the radii of the circle, so we mark the radius\r\n" );
document.write( "at the bottom left 2x.  That means that each side of the\r\n" );
document.write( "hexagon is the same length as the radius.\r\n" );
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document.write( " So we only need to look at this one triangle at the bottom.  I'll\r\n" );
document.write( "erase everything but it.\r\n" );
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document.write( "We can use the Pythagorean theorem on that right triangle\r\n" );
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document.write( "   c² = a² + b²\r\n" );
document.write( "(2x)² = x² + 4.8²\r\n" );
document.write( "  4x² = x² + 23.04  \r\n" );
document.write( "  3x² = 23.04\r\n" );
document.write( "   x² = 7.68\r\n" );
document.write( "    x = √7.68\r\n" );
document.write( "    x = 2.771281292 cm\r\n" );
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document.write( "So each side of the hexagon is 2x or 2(2.771281292) = 5.542562584 cm,\r\n" );
document.write( "and 2x is also the radius of the circle.\r\n" );
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document.write( "Answer: The side and the radius are both 5.542562584 cm.\r\n" );
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document.write( "The perimeter is 6 times the length of side, so the perimeter is\r\n" );
document.write( "6(5.542562584) = 33.25537551 cm.\r\n" );
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document.write( "Edwin

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