83 lines
5.2 KiB
HTML
83 lines
5.2 KiB
HTML
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<h2>Spherical triangles</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 332</h3><span style="width:300px;color:#666;">Published on Sunday, 10th April 2011, 10:00 am; Solved by 350</span></div>
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<p>A <b>spherical triangle</b> is a figure formed on the surface of a sphere by three <b>great circular arcs</b> intersecting pairwise in three vertices.</p>
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<div align='center'><img src="project/images/p_332_spherical.jpg" /></div>
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<p>Let C(<var>r</var>) be the sphere with the centre (0,0,0) and radius <var>r</var>.<br />
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Let Z(<var>r</var>) be the set of points on the surface of C(<var>r</var>) with integer coordinates.<br />
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Let T(<var>r</var>) be the set of spherical triangles with vertices in Z(<var>r</var>).
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Degenerate spherical triangles, formed by three points on the same great arc, are <u>not</u> included in T(<var>r</var>).<br />
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Let A(<var>r</var>) be the area of the smallest spherical triangle in T(<var>r</var>).</p>
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<p>For example A(14) is 3.294040 rounded to six decimal places.</p>
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<p>Find <img style="vertical-align:middle" src="project/images/p_332_sum.gif" /> A(<var>r</var>). Give your answer rounded to six decimal places.</p>
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