Lesson A problem on three spheres
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<H2>A problem on three spheres</H2> <H3>Problem 1</H3>The center of each of three spheres of radius R lies in the surfaces of the other two. Pass a plane containing the centers of the spheres. Find the area common to the three great circles cut from the spheres by this plane. <B>Solution</B> <pre> The centers of the three great circles form an equilateral triangle. The sides of this triangle are of the length R, since R is the radius of each of the three spheres. The area of this triangle is a = {{{(sqrt(3)/4)*R^2}}}. But the common intersection area is {{{highlight(highlight(WIDER))}}} than this triangle. The common area is formed as the intersection of three {{{highlight(highlight(sectors))}}} of the great circles, each sector with the central angle of 60°. So, the common area can be described as the equilateral triangle with the sides length of R {{{highlight(highlight(PLUS))}}} three adjacent circular <U>SEGMENTS</U> adjacent to each side of this triangle. The area of each such a segment is b = {{{(1/6)*pi*R^2}}} - {{{(sqrt(3)/4)*R^2}}}. The area of the three such segments is 3b = {{{(pi/2)*R^2}}} - {{{((3*sqrt(3))/4)*R^2}}}. Therefore, the total common area of the intersection of the three great circles is Area = a + 3b = {{{(sqrt(3)/4)*R^2}}} + {{{(pi/2)*R^2}}} - {{{((3*sqrt(3))/4)*R^2}}} = = {{{(pi/2)*R^2}}} - {{{((2*sqrt(3))/4)*R^2}}} = {{{(pi/2)*R^2}}} - {{{((sqrt(3))/2)*R^2}}} = = {{{((pi/2) - sqrt(3)/2)*R^2}}}. <U>ANSWER</U>. The common area of intersection of the three great circles is {{{((pi/2) - sqrt(3)/2)*R^2}}}, or about 0.6998*R^2, approximately. </pre> My other additional lessons on miscellaneous Geometry problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Nice-geometry-problem-of-a-Math-Olympiad-level.lesson>Nice geometry problem of a Math Olympiad level</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/OVERVIEW-of-my-lessons-on-additional-misc-Geometry-problems.lesson>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</A> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
