document.write( "Question 1208232: what is the largest diameter circle that will fit inside a regular pentagon with 2-inch sides
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #846516 by ikleyn(52777) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "what is the $\"highlight%28cross%28largest%29%29\"$ diameter of the circle that will fit inside a regular pentagon with 2-inch sides
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I crossed above, since there is NO the largest diameter.
\n" ); document.write( " All diameters of a circle have the same length.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Consider one of 5 isosceles triangles that make this pentagon.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The base of this triangle is 2 inches, hence, half of the base is 1 inch long.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The angle of the triangle, opposite to the base, is  360/5 = 72 degrees;\r\n" );
document.write( "half of this angle is 36 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let r be the radius of the circle, inscribed in this pentagon (in inches).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then   = tan(36).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence,  r =  =  = 1.3763819 inches.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For tan(36), there is special formula  tan(36) = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For deriving this formula, see, for example, this source\r\n" );
document.write( "\r\n" );
document.write( "    https://www.cuemath.com/trigonometry/tan-36-degrees/\r\n" );
document.write( "\r\n" );
document.write( "or many others (it is a classic in Trigonometry).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the answer to your question for the diameter is  \r\n" );
document.write( "\r\n" );
document.write( "    D =  =  =  2.752763841 inches.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );