document.write( "Question 1071351: I have Three Equal Inscribed Circles Into Big Circle. Diameter of internal circles is 5 mm. How I calculate area of outer circle. \n" ); document.write( "

Algebra.Com's Answer #686322 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Let me re-formulate your problem to make it more accurate.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "     Three Equal Circles  are Inscribed Into Big Circle in a way they touch each other. \r\n" );
document.write( "     Diameter of internal circles is 5 mm. Calculate the area of outer circle.\r\n" );
document.write( "

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

\r\n" );
document.write( "Let the points A, B and C be the centers of the three small circles. \r\n" );
document.write( "Draw the segments AB, AC and BC. \r\n" );
document.write( "You will get an equilateral triangle ABC, obviously.\r\n" );
document.write( "\r\n" );
document.write( "The sides of this triangle have the length twice the radius of the small circles, i.e. 10 mm. Obviously.\r\n" );
document.write( "\r\n" );
document.write( "In other words, the side length of this equilateral triangle is a = D = 5 mm.\r\n" );
document.write( "\r\n" );
document.write( "The length of any altitude of this equilateral triangle is  h = . (well known fact, actually) \r\n" );
document.write( "\r\n" );
document.write( "From the symmetry, it is clear that the intersection point of all three altitudes of the triangle ABC coincides with the center \r\n" );
document.write( "of the large circle O. \r\n" );
document.write( "\r\n" );
document.write( "Since the three altitudes of the triangle ABC coincide with its medians, the intersection point O divides each altitude/median\r\n" );
document.write( "in proportion 2:1, counting from the vertex to the base.\r\n" );
document.write( "\r\n" );
document.write( "Therefore, each of the three segments OA, OB and OC has the length  = .\r\n" );
document.write( "\r\n" );
document.write( "Then the radius of the large circle is  R = |OA| +  = .\r\n" );
document.write( "\r\n" );
document.write( "Now you can calculate R on your own by substituting a = 5 mm into the last formula, and then calculate the area of the large circle A = .\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" );