document.write( "Question 1209347: We cut a regular octagon ABCDEFGH out of a piece of cardboard. If $AB = 1$, then what is the area of the octagon?
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Algebra.Com's Answer #848585 by ikleyn(52776) $\"\"$ $\"About$

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\n" ); document.write( "We cut a regular octagon ABCDEFGH out of a piece of cardboard.
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\n" ); document.write( "\n" ); document.write( " Here is another way to solve the problem.\r
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document.write( "Let R be the radius of the circumscribed circle around our octagon.\r\n" );
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document.write( "Let's find its radius via the side length s.\r\n" );
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document.write( "The octagon consists of 8 congruent isosceles triangles, having \r\n" );
document.write( "the common vertex in the center.\r\n" );
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document.write( "Each such a triangle is an isosceles triangle with the lateral sides of the length R\r\n" );
document.write( "and the angle at the vertex of 45°.  Write the cosine rule equation for such a triangle \r\n" );
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document.write( "     =  +  - ,\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "     = .    (1)\r\n" );
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document.write( "In our case with s = 1, the last formula takes the form\r\n" );
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document.write( "     =  =  =  =  =  = .    (2)\r\n" );
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document.write( "Now the area of one such a triangle is\r\n" );
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document.write( "     =  =  =  =  = .    (3)\r\n" );
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document.write( "For the area of the entire octagon, we should take the quantity (3)  8 (eight) times to get\r\n" );
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document.write( "     =   square units.    ANSWER\r\n" );
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